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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere</div> <p>In mathematics, the <b>uniformization theorem</b> states that every <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a> is <a href="/wiki/Conformal_equivalence" class="mw-redirect" title="Conformal equivalence">conformally equivalent</a> to one of three Riemann surfaces: the open <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a>, the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, or the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a>. The theorem is a generalization of the <a href="/wiki/Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a> from simply connected <a href="/wiki/Open_set" title="Open set">open</a> <a href="/wiki/Subset" title="Subset">subsets</a> of the plane to arbitrary simply connected Riemann surfaces. </p><p>Since every Riemann surface has a <a href="/wiki/Universal_Cover" class="mw-redirect" title="Universal Cover">universal cover</a> which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> of <a href="/wiki/Constant_curvature" title="Constant curvature">constant curvature</a>, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. </p><p>The uniformization theorem also yields a similar classification of closed <a href="/wiki/Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable</a> <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian 2-manifolds</a> into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Felix <a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a> (<a href="#CITEREFKlein1883">1883</a>) and Henri <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> (<a href="#CITEREFPoincaré1882">1882</a>) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Henri Poincaré (<a href="#CITEREFPoincaré1883">1883</a>) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> (<a href="#CITEREFPoincaré1907">1907</a>) and <a href="/wiki/Paul_Koebe" title="Paul Koebe">Paul Koebe</a> (<a href="#CITEREFKoebe1907a">1907a</a>, <a href="#CITEREFKoebe1907b">1907b</a>, <a href="#CITEREFKoebe1907c">1907c</a>). Paul Koebe later gave several more proofs and generalizations. The history is described in <a href="#CITEREFGray1994">Gray (1994)</a>; a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in <a href="#CITEREFde_Saint-Gervais2016">de Saint-Gervais (2016)</a> (the <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a>-type pseudonym of the group of fifteen mathematicians who jointly produced this publication). </p> <div class="mw-heading mw-heading2"><h2 id="Classification_of_connected_Riemann_surfaces">Classification of connected Riemann surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=2" title="Edit section: Classification of connected Riemann surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surface</a> is the quotient of free, proper and holomorphic action of a <a href="/wiki/Discrete_group" title="Discrete group">discrete group</a> on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following: </p> <ol><li>the <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li>the complex plane</li> <li>the unit disk in the complex plane.</li></ol> <p>For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a> with fundamental group <span class="texhtml"><b>Z</b><sup>2</sup></span>; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_of_closed_oriented_Riemannian_2-manifolds">Classification of closed oriented Riemannian 2-manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=3" title="Edit section: Classification of closed oriented Riemannian 2-manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On an oriented 2-manifold, a <a href="/wiki/Riemannian_metric" class="mw-redirect" title="Riemannian metric">Riemannian metric</a> induces a complex structure using the passage to <a href="/wiki/Isothermal_coordinates" title="Isothermal coordinates">isothermal coordinates</a>. If the Riemannian metric is given locally as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>E</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>F</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>G</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102416b229bdb308773ed725bb8c4e62d8eb9a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.788ex; height:3.009ex;" alt="{\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}"></span></dd></dl> <p>then in the complex coordinate <i>z</i> = <i>x</i> + i<i>y</i>, it takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=\lambda |dz+\mu \,d{\overline {z}}|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>z</mi> <mo>+</mo> <mi>μ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=\lambda |dz+\mu \,d{\overline {z}}|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae1345e08eda40dcecac5bb9cd48f648acd4eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.166ex; height:3.343ex;" alt="{\displaystyle ds^{2}=\lambda |dz+\mu \,d{\overline {z}}|^{2},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {1}{4}}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\ \ \mu ={\frac {1}{4\lambda }}(E-G+2iF),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>E</mi> <mi>G</mi> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>G</mi> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {1}{4}}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\ \ \mu ={\frac {1}{4\lambda }}(E-G+2iF),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61e7879932c9363533454c81c5ad5d40727d62b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:57.577ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {1}{4}}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\ \ \mu ={\frac {1}{4\lambda }}(E-G+2iF),}"></span></dd></dl> <p>so that <i>λ</i> and <i>μ</i> are smooth with <i>λ</i> > 0 and |<i>μ</i>| < 1. In isothermal coordinates (<i>u</i>, <i>v</i>) the metric should take the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=\rho (du^{2}+dv^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=\rho (du^{2}+dv^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181d64f0472932874ef218f6dd69f7d92675870a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.308ex; height:3.176ex;" alt="{\displaystyle ds^{2}=\rho (du^{2}+dv^{2})}"></span></dd></dl> <p>with <i>ρ</i> > 0 smooth. The complex coordinate <i>w</i> = <i>u</i> + i <i>v</i> satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \,|dw|^{2}=\rho |w_{z}|^{2}\left|dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}\right|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>w</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>|</mo> <mrow> <mi>d</mi> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \,|dw|^{2}=\rho |w_{z}|^{2}\left|dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}\right|^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f3921c73e12d50f3b6f74740eaa34ee6cf63cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.679ex; height:6.009ex;" alt="{\displaystyle \rho \,|dw|^{2}=\rho |w_{z}|^{2}\left|dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}\right|^{2},}"></span></dd></dl> <p>so that the coordinates (<i>u</i>, <i>v</i>) will be isothermal locally provided the <a href="/wiki/Beltrami_equation" title="Beltrami equation">Beltrami equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>w</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/939f41ded8aad882467f2431e21cfc31641576fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.137ex; height:5.676ex;" alt="{\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}}"></span></dd></dl> <p>has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian. </p><p>These conditions can be phrased equivalently in terms of the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> and the <a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a> <span class="texhtml">∗</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="texhtml"><i>u</i></span> and <span class="texhtml"><i>v</i></span> will be isothermal coordinates if <span class="texhtml">∗<i>du</i> = <i>dv</i></span>, where <span class="texhtml">∗</span> is defined on differentials by <span class="texhtml">∗(<i>p</i> <i>dx</i> + <i>q</i> <i>dy</i>) = −<i>q</i> <i>dx</i> + <i>p</i> <i>dy</i></span>. Let <span class="texhtml">∆ = ∗<i>d</i>∗<i>d</i></span> be the <a href="/wiki/Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a>. By standard elliptic theory, <span class="texhtml"><i>u</i></span> can be chosen to be <a href="/wiki/Harmonic" title="Harmonic">harmonic</a> near a given point, i.e. <span class="texhtml">Δ <i>u</i> = 0</span>, with <span class="texhtml"><i>du</i></span> non-vanishing. By the <a href="/wiki/Poincar%C3%A9_lemma" title="Poincaré lemma">Poincaré lemma</a> <span class="texhtml"><i>dv</i> = ∗<i>du</i></span> has a local solution <span class="texhtml"><i>v</i></span> exactly when <span class="texhtml"><i>d</i>(∗<i>du</i>) = 0</span>. This condition is equivalent to <span class="texhtml">Δ <i>u</i> = 0</span>, so can always be solved locally. Since <span class="texhtml"><i>du</i></span> is non-zero and the square of the Hodge star operator is −1 on 1-forms, <span class="texhtml"><i>du</i></span> and <span class="texhtml"><i>dv</i></span> must be linearly independent, so that <span class="texhtml"><i>u</i></span> and <span class="texhtml"><i>v</i></span> give local isothermal coordinates. </p><p>The existence of isothermal coordinates can be proved by other methods, for example using the <a href="/wiki/Beltrami_equation#Solution_in_L2_for_smooth_Beltrami_coefficients" title="Beltrami equation">general theory of the Beltrami equation</a>, as in <a href="#CITEREFAhlfors2006">Ahlfors (2006)</a>, or by direct elementary methods, as in <a href="#CITEREFChern1955">Chern (1955)</a> and <a href="#CITEREFJost2006">Jost (2006)</a>. </p><p>From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of <a href="/wiki/Constant_curvature" title="Constant curvature">constant curvature</a>, so a <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient</a> of one of the following by a <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">free action</a> of a <a href="/wiki/Discrete_group" title="Discrete group">discrete subgroup</a> of an <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a>: </p> <ol><li>the <a href="/wiki/Sphere" title="Sphere">sphere</a> (curvature +1)</li> <li>the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> (curvature 0)</li> <li>the <a href="/wiki/Hyperbolic_space" title="Hyperbolic space">hyperbolic plane</a> (curvature −1).</li></ol> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Orange_Sphere.png" class="mw-file-description" title="genus 0"><img alt="genus 0" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Orange_Sphere.png/114px-Orange_Sphere.png" decoding="async" width="114" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Orange_Sphere.png/172px-Orange_Sphere.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Orange_Sphere.png/229px-Orange_Sphere.png 2x" data-file-width="800" data-file-height="839" /></a></span></div> <div class="gallerytext">genus 0</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Orange_Torus.png" class="mw-file-description" title="genus 1"><img alt="genus 1" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Orange_Torus.png/120px-Orange_Torus.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Orange_Torus.png/180px-Orange_Torus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Orange_Torus.png/240px-Orange_Torus.png 2x" data-file-width="800" data-file-height="776" /></a></span></div> <div class="gallerytext">genus 1</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Orange_Genus_2_Surface.png" class="mw-file-description" title="genus 2"><img alt="genus 2" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Orange_Genus_2_Surface.png/120px-Orange_Genus_2_Surface.png" decoding="async" width="120" height="42" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Orange_Genus_2_Surface.png/180px-Orange_Genus_2_Surface.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Orange_Genus_2_Surface.png/240px-Orange_Genus_2_Surface.png 2x" data-file-width="1497" data-file-height="520" /></a></span></div> <div class="gallerytext">genus 2</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Orange_Genus_3_Surface.png" class="mw-file-description" title="genus 3"><img alt="genus 3" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Orange_Genus_3_Surface.png/111px-Orange_Genus_3_Surface.png" decoding="async" width="111" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Orange_Genus_3_Surface.png/167px-Orange_Genus_3_Surface.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Orange_Genus_3_Surface.png/222px-Orange_Genus_3_Surface.png 2x" data-file-width="1093" data-file-height="1180" /></a></span></div> <div class="gallerytext">genus 3</div> </li> </ul> <p>The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> (equal to 2). The second gives all flat 2-manifolds, i.e. the <a href="/wiki/Torus" title="Torus">tori</a>, which have Euler characteristic 0. The third case covers all 2-manifolds of constant negative curvature, i.e. the <i>hyperbolic</i> 2-manifolds all of which have negative Euler characteristic. The classification is consistent with the <a href="/wiki/Gauss%E2%80%93Bonnet_theorem" title="Gauss–Bonnet theorem">Gauss–Bonnet theorem</a>, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2<i>g</i>, where <i>g</i> is the genus of the 2-manifold, i.e. the number of "holes". </p> <div class="mw-heading mw-heading2"><h2 id="Methods_of_proof">Methods of proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=4" title="Edit section: Methods of proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many classical proofs of the uniformization theorem rely on constructing a real-valued <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic function</a> on the simply connected Riemann surface, possibly with a singularity at one or two points and often corresponding to a form of <a href="/wiki/Green%27s_function" title="Green's function">Green's function</a>. Four methods of constructing the harmonic function are widely employed: the <a href="/wiki/Perron_method" title="Perron method">Perron method</a>; the <a href="/wiki/Schwarz_alternating_method" title="Schwarz alternating method">Schwarz alternating method</a>; <a href="/wiki/Dirichlet%27s_principle" title="Dirichlet's principle">Dirichlet's principle</a>; and <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a>'s method of orthogonal projection. In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics. These include the <a href="/wiki/Beltrami_equation" title="Beltrami equation">Beltrami equation</a> from <a href="/wiki/Teichm%C3%BCller_theory" class="mw-redirect" title="Teichmüller theory">Teichmüller theory</a> and an equivalent formulation in terms of <a href="/wiki/Harmonic_map" title="Harmonic map">harmonic maps</a>; <a href="/wiki/Liouville%27s_equation" title="Liouville's equation">Liouville's equation</a>, already studied by Poincaré; and <a href="/wiki/Ricci_flow" title="Ricci flow">Ricci flow</a> along with other nonlinear flows. </p><p><a href="/wiki/Rad%C3%B3%27s_theorem_(Riemann_surfaces)" title="Radó's theorem (Riemann surfaces)">Rado's theorem</a> shows that every Riemann surface is automatically <a href="/wiki/Second-countable_space" title="Second-countable space">second-countable</a>. Although Rado's theorem is often used in proofs of the uniformization theorem, some proofs have been formulated so that Rado's theorem becomes a consequence. Second countability is automatic for compact Riemann surfaces. </p> <div class="mw-heading mw-heading3"><h3 id="Hilbert_space_methods">Hilbert space methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=5" title="Edit section: Hilbert space methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Planar_Riemann_surface#Uniformization_theorem" title="Planar Riemann surface">Planar Riemann surface § Uniformization theorem</a></div> <p>In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>, the first edition incorporated <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert's</a> treatment of the <a href="/wiki/Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a> using <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> techniques; <a href="/wiki/L._E._J._Brouwer" title="L. E. J. Brouwer">Brouwer's</a> contributions to topology; and <a href="/wiki/Paul_Koebe" title="Paul Koebe">Koebe's</a> proof of the uniformization theorem and its subsequent improvements. Much later <a href="#CITEREFWeyl1940">Weyl (1940)</a> developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included <a href="/wiki/Weyl%27s_lemma_(Laplace_equation)" title="Weyl's lemma (Laplace equation)">Weyl's lemma</a> on <a href="/wiki/Elliptic_regularity" class="mw-redirect" title="Elliptic regularity">elliptic regularity</a>, was related to <a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge's</a> <a href="/wiki/Hodge_theory" title="Hodge theory">theory of harmonic integrals</a>; and both theories were subsumed into the modern theory of <a href="/wiki/Elliptic_operator" title="Elliptic operator">elliptic operators</a> and <span class="texhtml"><i>L</i><sup>2</sup></span> <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev spaces</a>. In the third edition of his book from 1955, translated into English in <a href="#CITEREFWeyl1964">Weyl (1964)</a>, Weyl adopted the modern definition of differential manifold, in preference to <a href="/wiki/Triangulation_(topology)" title="Triangulation (topology)">triangulations</a>, but decided not to make use of his method of orthogonal projection. <a href="#CITEREFSpringer1957">Springer (1957)</a> followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. <a href="#CITEREFKodaira2007">Kodaira (2007)</a> describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in <a href="#CITEREFDonaldson2011">Donaldson (2011)</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_flows">Nonlinear flows</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=6" title="Edit section: Nonlinear flows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Ricci_flow#Relationship_to_uniformization_and_geometrization" title="Ricci flow">Ricci flow § Relationship to uniformization and geometrization</a></div> <p><a href="/wiki/Richard_S._Hamilton" title="Richard S. Hamilton">Richard S. Hamilton</a> showed that the <a href="/wiki/Ricci_flow" title="Ricci flow">normalized Ricci flow</a> on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric). However, his proof relied on the uniformization theorem. The missing step involved Ricci flow on the 2-sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by <a href="#CITEREFChenLuTian2006">Chen, Lu & Tian (2006)</a>;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> a short self-contained account of Ricci flow on the 2-sphere was given in <a href="#CITEREFAndrewsBryan2010">Andrews & Bryan (2010)</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=7" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Koebe proved the <b>general uniformization theorem</b> that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere. </p><p>In 3 dimensions, there are 8 geometries, called the <a href="/wiki/Geometrization_conjecture#The_eight_Thurston_geometries" title="Geometrization conjecture">eight Thurston geometries</a>. Not every 3-manifold admits a geometry, but Thurston's <a href="/wiki/Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> proved by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> states that every 3-manifold can be cut into pieces that are geometrizable. </p><p>The <a href="/wiki/Simultaneous_uniformization_theorem" title="Simultaneous uniformization theorem">simultaneous uniformization theorem</a> of <a href="/wiki/Lipman_Bers" title="Lipman Bers">Lipman Bers</a> shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same <a href="/wiki/Quasi-Fuchsian_group" title="Quasi-Fuchsian group">quasi-Fuchsian group</a>. </p><p>The <a href="/wiki/Measurable_Riemann_mapping_theorem" title="Measurable Riemann mapping theorem">measurable Riemann mapping theorem</a> shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a <a href="/wiki/Quasiconformal_map" class="mw-redirect" title="Quasiconformal map">quasiconformal map</a> with any given bounded measurable Beltrami coefficient. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=P-adic_uniformization_theorem&action=edit&redlink=1" class="new" title="P-adic uniformization theorem (page does not exist)"><i>p</i>-adic uniformization theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFDeTurckKazdan1981">DeTurck & Kazdan 1981</a>; <a href="#CITEREFTaylor1996a">Taylor 1996a</a>, pp. 377–378</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrendle2010">Brendle 2010</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Historic_references">Historic references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=11" title="Edit section: Historic references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchwarz1870" class="citation cs2"><a href="/wiki/Hermann_Schwarz" title="Hermann Schwarz">Schwarz, H. A.</a> (1870), <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/34472#page/280/mode/1up">"Über einen Grenzübergang durch alternierendes Verfahren"</a>, <i>Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich</i>, <b>15</b>: 272–286, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:02.0214.02">02.0214.02</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Vierteljahrsschrift+der+Naturforschenden+Gesellschaft+in+Z%C3%BCrich&rft.atitle=%C3%9Cber+einen+Grenz%C3%BCbergang+durch+alternierendes+Verfahren&rft.volume=15&rft.pages=272-286&rft.date=1870&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A02.0214.02%23id-name%3DJFM&rft.aulast=Schwarz&rft.aufirst=H.+A.&rft_id=https%3A%2F%2Fwww.biodiversitylibrary.org%2Fitem%2F34472%23page%2F280%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1883" class="citation cs2">Klein, Felix (1883), <a rel="nofollow" class="external text" href="https://zenodo.org/record/2161412">"Neue Beiträge zur Riemann'schen Functionentheorie"</a>, <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>21</b> (2): 141–218, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01442920">10.1007/BF01442920</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:15.0351.01">15.0351.01</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120465625">120465625</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Neue+Beitr%C3%A4ge+zur+Riemann%27schen+Functionentheorie&rft.volume=21&rft.issue=2&rft.pages=141-218&rft.date=1883&rft.issn=0025-5831&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120465625%23id-name%3DS2CID&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A15.0351.01%23id-name%3DJFM&rft_id=info%3Adoi%2F10.1007%2FBF01442920&rft.aulast=Klein&rft.aufirst=Felix&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2161412&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoebe1907a" class="citation cs2">Koebe, P. (1907a), <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN00250118X">"Über die Uniformisierung reeller analytischer Kurven"</a>, <i>Göttinger Nachrichten</i>: 177–190, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:38.0453.01">38.0453.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.atitle=%C3%9Cber+die+Uniformisierung+reeller+analytischer+Kurven&rft.pages=177-190&rft.date=1907&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A38.0453.01%23id-name%3DJFM&rft.aulast=Koebe&rft.aufirst=P.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN00250118X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoebe1907b" class="citation cs2">Koebe, P. (1907b), <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002501198">"Über die Uniformisierung beliebiger analytischer Kurven"</a>, <i>Göttinger Nachrichten</i>: 191–210, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:38.0454.01">38.0454.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.atitle=%C3%9Cber+die+Uniformisierung+beliebiger+analytischer+Kurven&rft.pages=191-210&rft.date=1907&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A38.0454.01%23id-name%3DJFM&rft.aulast=Koebe&rft.aufirst=P.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN002501198&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoebe1907c" class="citation cs2">Koebe, P. (1907c), <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002501473">"Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung)"</a>, <i>Göttinger Nachrichten</i>: 633–669, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:38.0455.02">38.0455.02</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.atitle=%C3%9Cber+die+Uniformisierung+beliebiger+analytischer+Kurven+%28Zweite+Mitteilung%29&rft.pages=633-669&rft.date=1907&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A38.0455.02%23id-name%3DJFM&rft.aulast=Koebe&rft.aufirst=P.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN002501473&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoebe1910a" class="citation cs2">Koebe, Paul (1910a), "Über die Uniformisierung beliebiger analytischer Kurven", <i>Journal für die Reine und Angewandte Mathematik</i>, <b>138</b>: 192–253, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1910.138.192">10.1515/crll.1910.138.192</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120198686">120198686</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=%C3%9Cber+die+Uniformisierung+beliebiger+analytischer+Kurven&rft.volume=138&rft.pages=192-253&rft.date=1910&rft_id=info%3Adoi%2F10.1515%2Fcrll.1910.138.192&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120198686%23id-name%3DS2CID&rft.aulast=Koebe&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKoebe1910b" class="citation cs2">Koebe, Paul (1910b), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/pdfcache/PPN252457811_1910/PPN252457811_1910___LOG_0008.pdf">"Über die Hilbertsche Uniformlsierungsmethode"</a> <span class="cs1-format">(PDF)</span>, <i>Göttinger Nachrichten</i>: 61–65</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.atitle=%C3%9Cber+die+Hilbertsche+Uniformlsierungsmethode&rft.pages=61-65&rft.date=1910&rft.aulast=Koebe&rft.aufirst=Paul&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fpdfcache%2FPPN252457811_1910%2FPPN252457811_1910___LOG_0008.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1882" class="citation cs2">Poincaré, H. (1882), "Mémoire sur les fonctions fuchsiennes", <i><a href="/wiki/Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a></i>, <b>1</b>: 193–294, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02592135">10.1007/BF02592135</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-5962">0001-5962</a>, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:15.0342.01">15.0342.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=M%C3%A9moire+sur+les+fonctions+fuchsiennes&rft.volume=1&rft.pages=193-294&rft.date=1882&rft.issn=0001-5962&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A15.0342.01%23id-name%3DJFM&rft_id=info%3Adoi%2F10.1007%2FBF02592135&rft.aulast=Poincar%C3%A9&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1883" class="citation cs2"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré, Henri</a> (1883), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=BSMF_1883__11__112_1">"Sur un théorème de la théorie générale des fonctions"</a>, <i>Bulletin de la Société Mathématique de France</i>, <b>11</b>: 112–125, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.261">10.24033/bsmf.261</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0037-9484">0037-9484</a>, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:15.0348.01">15.0348.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+de+la+Soci%C3%A9t%C3%A9+Math%C3%A9matique+de+France&rft.atitle=Sur+un+th%C3%A9or%C3%A8me+de+la+th%C3%A9orie+g%C3%A9n%C3%A9rale+des+fonctions&rft.volume=11&rft.pages=112-125&rft.date=1883&rft.issn=0037-9484&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A15.0348.01%23id-name%3DJFM&rft_id=info%3Adoi%2F10.24033%2Fbsmf.261&rft.aulast=Poincar%C3%A9&rft.aufirst=Henri&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DBSMF_1883__11__112_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoincaré1907" class="citation cs2"><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré, Henri</a> (1907), <a rel="nofollow" class="external text" href="https://zenodo.org/records/2012595/files/article.pdf">"Sur l'uniformisation des fonctions analytiques"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a></i>, <b>31</b>: 1–63, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02415442">10.1007/BF02415442</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-5962">0001-5962</a>, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:38.0452.02">38.0452.02</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=Sur+l%27uniformisation+des+fonctions+analytiques&rft.volume=31&rft.pages=1-63&rft.date=1907&rft.issn=0001-5962&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A38.0452.02%23id-name%3DJFM&rft_id=info%3Adoi%2F10.1007%2FBF02415442&rft.aulast=Poincar%C3%A9&rft.aufirst=Henri&rft_id=https%3A%2F%2Fzenodo.org%2Frecords%2F2012595%2Ffiles%2Farticle.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1909" class="citation cs2">Hilbert, David (1909), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/pdfcache/PPN252457811_1909/PPN252457811_1909___LOG_0042.pdf">"Zur Theorie der konformen Abbildung"</a> <span class="cs1-format">(PDF)</span>, <i>Göttinger Nachrichten</i>: 314–323</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=G%C3%B6ttinger+Nachrichten&rft.atitle=Zur+Theorie+der+konformen+Abbildung&rft.pages=314-323&rft.date=1909&rft.aulast=Hilbert&rft.aufirst=David&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fpdfcache%2FPPN252457811_1909%2FPPN252457811_1909___LOG_0042.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerron1923" class="citation cs2"><a href="/wiki/Oskar_Perron" title="Oskar Perron">Perron, O.</a> (1923), "Eine neue Behandlung der ersten Randwertaufgabe für Δu=0", <i><a href="/wiki/Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i>, <b>18</b> (1): 42–54, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01192395">10.1007/BF01192395</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122843531">122843531</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Eine+neue+Behandlung+der+ersten+Randwertaufgabe+f%C3%BCr+%CE%94u%3D0&rft.volume=18&rft.issue=1&rft.pages=42-54&rft.date=1923&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122843531%23id-name%3DS2CID&rft.issn=0025-5874&rft_id=info%3Adoi%2F10.1007%2FBF01192395&rft.aulast=Perron&rft.aufirst=O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1913" class="citation cs2">Weyl, Hermann (1913), <i>Die Idee der Riemannschen Fläche (1997 reprint of the 1913 German original)</i>, Teubner, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-8154-2096-6" title="Special:BookSources/978-3-8154-2096-6"><bdi>978-3-8154-2096-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Die+Idee+der+Riemannschen+Fl%C3%A4che+%281997+reprint+of+the+1913+German+original%29&rft.pub=Teubner&rft.date=1913&rft.isbn=978-3-8154-2096-6&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1940" class="citation cs2">Weyl, Hermann (1940), "The method of orthogonal projections in potential theory", <i>Duke Math. J.</i>, <b>7</b>: 411–444, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fs0012-7094-40-00725-6">10.1215/s0012-7094-40-00725-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Math.+J.&rft.atitle=The+method+of+orthogonal+projections+in+potential+theory&rft.volume=7&rft.pages=411-444&rft.date=1940&rft_id=info%3Adoi%2F10.1215%2Fs0012-7094-40-00725-6&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical_surveys">Historical surveys</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=12" title="Edit section: Historical surveys"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbikoff1981" class="citation cs2">Abikoff, William (1981), "The uniformization theorem", <i>Amer. Math. Monthly</i>, <b>88</b> (8): 574–592, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2320507">10.2307/2320507</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2320507">2320507</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Amer.+Math.+Monthly&rft.atitle=The+uniformization+theorem&rft.volume=88&rft.issue=8&rft.pages=574-592&rft.date=1981&rft_id=info%3Adoi%2F10.2307%2F2320507&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2320507%23id-name%3DJSTOR&rft.aulast=Abikoff&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGray1994" class="citation cs2">Gray, Jeremy (1994), <a rel="nofollow" class="external text" href="http://www.math.stonybrook.edu/~bishop/classes/math401.F09/GrayRMT.pdf">"On the history of the Riemann mapping theorem"</a> <span class="cs1-format">(PDF)</span>, <i>Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento</i> (34): 47–94, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1295591">1295591</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti+del+Circolo+Matematico+di+Palermo.+Serie+II.+Supplemento&rft.atitle=On+the+history+of+the+Riemann+mapping+theorem&rft.issue=34&rft.pages=47-94&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1295591%23id-name%3DMR&rft.aulast=Gray&rft.aufirst=Jeremy&rft_id=http%3A%2F%2Fwww.math.stonybrook.edu%2F~bishop%2Fclasses%2Fmath401.F09%2FGrayRMT.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBottazziniGray2013" class="citation cs2">Bottazzini, Umberto; Gray, Jeremy (2013), <i>Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory</i>, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1461457251" title="Special:BookSources/978-1461457251"><bdi>978-1461457251</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hidden+Harmony%E2%80%94Geometric+Fantasies%3A+The+Rise+of+Complex+Function+Theory&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer&rft.date=2013&rft.isbn=978-1461457251&rft.aulast=Bottazzini&rft.aufirst=Umberto&rft.au=Gray%2C+Jeremy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Saint-Gervais2016" class="citation cs2">de Saint-Gervais, Henri Paul (2016), <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.ems-ph.org/books/book.php?proj_nr=198"><i>Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem</i></a></span>, translated by Robert G. Burns, European Mathematical Society, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2F145">10.4171/145</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-145-3" title="Special:BookSources/978-3-03719-145-3"><bdi>978-3-03719-145-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Uniformization+of+Riemann+Surfaces%3A+revisiting+a+hundred-year-old+theorem&rft.pub=European+Mathematical+Society&rft.date=2016&rft_id=info%3Adoi%2F10.4171%2F145&rft.isbn=978-3-03719-145-3&rft.aulast=de+Saint-Gervais&rft.aufirst=Henri+Paul&rft_id=http%3A%2F%2Fwww.ems-ph.org%2Fbooks%2Fbook.php%3Fproj_nr%3D198&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span>, translation of <a rel="nofollow" class="external text" href="http://perso.ens-lyon.fr/ghys/articles/Uniformisationsurfaces.pdf">French text</a> (prepared in 2007 during centenary of 1907 papers of Koebe and Poincaré)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Harmonic_functions">Harmonic functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=13" title="Edit section: Harmonic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Perron's method</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeins1949" class="citation cs2">Heins, M. (1949), "The conformal mapping of simply-connected Riemann surfaces", <i>Ann. of Math.</i>, <b>50</b> (3): 686–690, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969555">10.2307/1969555</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969555">1969555</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+of+Math.&rft.atitle=The+conformal+mapping+of+simply-connected+Riemann+surfaces&rft.volume=50&rft.issue=3&rft.pages=686-690&rft.date=1949&rft_id=info%3Adoi%2F10.2307%2F1969555&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969555%23id-name%3DJSTOR&rft.aulast=Heins&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeins1951" class="citation cs2">Heins, M. (1951), "Interior mapping of an orientable surface into <i>S</i><sup>2</sup>", <i>Proc. Amer. Math. Soc.</i>, <b>2</b> (6): 951–952, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9939-1951-0045221-4">10.1090/s0002-9939-1951-0045221-4</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Amer.+Math.+Soc.&rft.atitle=Interior+mapping+of+an+orientable+surface+into+S%3Csup%3E2%3C%2Fsup%3E&rft.volume=2&rft.issue=6&rft.pages=951-952&rft.date=1951&rft_id=info%3Adoi%2F10.1090%2Fs0002-9939-1951-0045221-4&rft.aulast=Heins&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeins1957" class="citation cs2">Heins, M. (1957), <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-12/issue-none/The-conformal-mapping-of-simply-connected-Riemann-surfaces-II/nmj/1118799932.pdf">"The conformal mapping of simply-connected Riemann surfaces. II"</a> <span class="cs1-format">(PDF)</span>, <i>Nagoya Math. J.</i>, <b>12</b>: 139–143, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs002776300002198x">10.1017/s002776300002198x</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nagoya+Math.+J.&rft.atitle=The+conformal+mapping+of+simply-connected+Riemann+surfaces.+II&rft.volume=12&rft.pages=139-143&rft.date=1957&rft_id=info%3Adoi%2F10.1017%2Fs002776300002198x&rft.aulast=Heins&rft.aufirst=M.&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fnagoya-mathematical-journal%2Fvolume-12%2Fissue-none%2FThe-conformal-mapping-of-simply-connected-Riemann-surfaces-II%2Fnmj%2F1118799932.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfluger1957" class="citation cs2">Pfluger, Albert (1957), <i>Theorie der Riemannschen Flächen</i>, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+der+Riemannschen+Fl%C3%A4chen&rft.pub=Springer&rft.date=1957&rft.aulast=Pfluger&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlfors2010" class="citation cs2">Ahlfors, Lars V. (2010), <i>Conformal invariants: topics in geometric function theory</i>, AMS Chelsea Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-5270-5" title="Special:BookSources/978-0-8218-5270-5"><bdi>978-0-8218-5270-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conformal+invariants%3A+topics+in+geometric+function+theory&rft.pub=AMS+Chelsea+Publishing&rft.date=2010&rft.isbn=978-0-8218-5270-5&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeardon1984" class="citation cs2">Beardon, A. F. (1984), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/primeronriemanns0000bear">"A primer on Riemann surfaces"</a></span>, <i>London Mathematical Society Lecture Note Series</i>, <b>78</b>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521271042" title="Special:BookSources/978-0521271042"><bdi>978-0521271042</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=London+Mathematical+Society+Lecture+Note+Series&rft.atitle=A+primer+on+Riemann+surfaces&rft.volume=78&rft.date=1984&rft.isbn=978-0521271042&rft.aulast=Beardon&rft.aufirst=A.+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprimeronriemanns0000bear&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFForster1991" class="citation cs2">Forster, Otto (1991), <i>Lectures on Riemann surfaces</i>, Graduate Texts in Mathematics, vol. 81, translated by Bruce Gilligan, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90617-1" title="Special:BookSources/978-0-387-90617-1"><bdi>978-0-387-90617-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Riemann+surfaces&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-90617-1&rft.aulast=Forster&rft.aufirst=Otto&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFarkasKra1980" class="citation cs2">Farkas, Hershel M.; Kra, Irwin (1980), <i>Riemann surfaces</i> (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90465-8" title="Special:BookSources/978-0-387-90465-8"><bdi>978-0-387-90465-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann+surfaces&rft.edition=2nd&rft.pub=Springer&rft.date=1980&rft.isbn=978-0-387-90465-8&rft.aulast=Farkas&rft.aufirst=Hershel+M.&rft.au=Kra%2C+Irwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGamelin2001" class="citation cs2">Gamelin, Theodore W. (2001), <i>Complex analysis</i>, Undergraduate Texts in Mathematics, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95069-3" title="Special:BookSources/978-0-387-95069-3"><bdi>978-0-387-95069-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-387-95069-3&rft.aulast=Gamelin&rft.aufirst=Theodore+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHubbard2006" class="citation cs2">Hubbard, John H. (2006), <i>Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1. Teichmüller theory</i>, Matrix Editions, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0971576629" title="Special:BookSources/978-0971576629"><bdi>978-0971576629</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Teichm%C3%BCller+theory+and+applications+to+geometry%2C+topology%2C+and+dynamics.+Vol.+1.+Teichm%C3%BCller+theory&rft.pub=Matrix+Editions&rft.date=2006&rft.isbn=978-0971576629&rft.aulast=Hubbard&rft.aufirst=John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlag2014" class="citation cs2">Schlag, Wilhelm (2014), <i>A course in complex analysis and Riemann surfaces.</i>, Graduate Studies in Mathematics, vol. 154, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-9847-5" title="Special:BookSources/978-0-8218-9847-5"><bdi>978-0-8218-9847-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+course+in+complex+analysis+and+Riemann+surfaces.&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2014&rft.isbn=978-0-8218-9847-5&rft.aulast=Schlag&rft.aufirst=Wilhelm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Schwarz's alternating method</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNevanlinna1953" class="citation cs2">Nevanlinna, Rolf (1953), <i>Uniformisierung</i>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 64, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-52801-9">10.1007/978-3-642-52801-9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-52802-6" title="Special:BookSources/978-3-642-52802-6"><bdi>978-3-642-52802-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Uniformisierung&rft.series=Die+Grundlehren+der+Mathematischen+Wissenschaften+in+Einzeldarstellungen+mit+besonderer+Ber%C3%BCcksichtigung+der+Anwendungsgebiete&rft.pub=Springer&rft.date=1953&rft_id=info%3Adoi%2F10.1007%2F978-3-642-52801-9&rft.isbn=978-3-642-52802-6&rft.aulast=Nevanlinna&rft.aufirst=Rolf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBehnkeSommer1965" class="citation cs2">Behnke, Heinrich; Sommer, Friedrich (1965), <i>Theorie der analytischen Funktionen einer komplexen Veränderlichen</i>, Die Grundlehren der mathematischen Wissenschaften, vol. 77 (3rd ed.), Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+der+analytischen+Funktionen+einer+komplexen+Ver%C3%A4nderlichen&rft.series=Die+Grundlehren+der+mathematischen+Wissenschaften&rft.edition=3rd&rft.pub=Springer&rft.date=1965&rft.aulast=Behnke&rft.aufirst=Heinrich&rft.au=Sommer%2C+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreitag2011" class="citation cs2">Freitag, Eberhard (2011), <i>Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-20553-8" title="Special:BookSources/978-3-642-20553-8"><bdi>978-3-642-20553-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis.+2.+Riemann+surfaces%2C+several+complex+variables%2C+abelian+functions%2C+higher+modular+functions&rft.pub=Springer&rft.date=2011&rft.isbn=978-3-642-20553-8&rft.aulast=Freitag&rft.aufirst=Eberhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Dirichlet principle</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1964" class="citation cs2">Weyl, Hermann (1964), <i>The concept of a Riemann surface</i>, translated by Gerald R. MacLane, Addison-Wesley, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0069903">0069903</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+concept+of+a+Riemann+surface&rft.pub=Addison-Wesley&rft.date=1964&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0069903%23id-name%3DMR&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourant1977" class="citation cs2">Courant, Richard (1977), <i>Dirichlet's principle, conformal mapping, and minimal surfaces</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90246-3" title="Special:BookSources/978-0-387-90246-3"><bdi>978-0-387-90246-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dirichlet%27s+principle%2C+conformal+mapping%2C+and+minimal+surfaces&rft.pub=Springer&rft.date=1977&rft.isbn=978-0-387-90246-3&rft.aulast=Courant&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegel1988" class="citation cs2">Siegel, C. L. (1988), <i>Topics in complex function theory. Vol. I. Elliptic functions and uniformization theory</i>, translated by A. Shenitzer; D. Solitar, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471608448" title="Special:BookSources/978-0471608448"><bdi>978-0471608448</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+complex+function+theory.+Vol.+I.+Elliptic+functions+and+uniformization+theory&rft.pub=Wiley&rft.date=1988&rft.isbn=978-0471608448&rft.aulast=Siegel&rft.aufirst=C.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Weyl's method of orthogonal projection</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpringer1957" class="citation cs2">Springer, George (1957), <i>Introduction to Riemann surfaces</i>, Addison-Wesley, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0092855">0092855</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Riemann+surfaces&rft.pub=Addison-Wesley&rft.date=1957&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0092855%23id-name%3DMR&rft.aulast=Springer&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKodaira2007" class="citation cs2">Kodaira, Kunihiko (2007), <i>Complex analysis</i>, Cambridge Studies in Advanced Mathematics, vol. 107, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521809375" title="Special:BookSources/9780521809375"><bdi>9780521809375</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=9780521809375&rft.aulast=Kodaira&rft.aufirst=Kunihiko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonaldson2011" class="citation cs2">Donaldson, Simon (2011), <i>Riemann surfaces</i>, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-960674-0" title="Special:BookSources/978-0-19-960674-0"><bdi>978-0-19-960674-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann+surfaces&rft.series=Oxford+Graduate+Texts+in+Mathematics&rft.pub=Oxford+University+Press&rft.date=2011&rft.isbn=978-0-19-960674-0&rft.aulast=Donaldson&rft.aufirst=Simon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Sario operators</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSario1952" class="citation cs2">Sario, Leo (1952), "A linear operator method on arbitrary Riemann surfaces", <i>Trans. Amer. Math. Soc.</i>, <b>72</b> (2): 281–295, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1952-0046442-2">10.1090/s0002-9947-1952-0046442-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=A+linear+operator+method+on+arbitrary+Riemann+surfaces&rft.volume=72&rft.issue=2&rft.pages=281-295&rft.date=1952&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1952-0046442-2&rft.aulast=Sario&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlforsSario1960" class="citation cs2">Ahlfors, Lars V.; Sario, Leo (1960), <i>Riemann surfaces</i>, Princeton Mathematical Series, vol. 26, Princeton University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Riemann+surfaces&rft.series=Princeton+Mathematical+Series&rft.pub=Princeton+University+Press&rft.date=1960&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rft.au=Sario%2C+Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_differential_equations">Nonlinear differential equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniformization_theorem&action=edit&section=14" title="Edit section: Nonlinear differential equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Beltrami's equation</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlfors2006" class="citation cs2">Ahlfors, Lars V. (2006), <i>Lectures on quasiconformal mappings</i>, University Lecture Series, vol. 38 (2nd ed.), American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3644-6" title="Special:BookSources/978-0-8218-3644-6"><bdi>978-0-8218-3644-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+quasiconformal+mappings&rft.series=University+Lecture+Series&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-0-8218-3644-6&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhlforsBers1960" class="citation cs2">Ahlfors, Lars V.; Bers, Lipman (1960), "Riemann's mapping theorem for variable metrics", <i>Ann. of Math.</i>, <b>72</b> (2): 385–404, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970141">10.2307/1970141</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970141">1970141</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+of+Math.&rft.atitle=Riemann%27s+mapping+theorem+for+variable+metrics&rft.volume=72&rft.issue=2&rft.pages=385-404&rft.date=1960&rft_id=info%3Adoi%2F10.2307%2F1970141&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970141%23id-name%3DJSTOR&rft.aulast=Ahlfors&rft.aufirst=Lars+V.&rft.au=Bers%2C+Lipman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBers1960" class="citation cs2">Bers, Lipman (1960), <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-66/issue-2/Simultaneous-uniformization/bams/1183523461.pdf">"Simultaneous uniformization"</a> <span class="cs1-format">(PDF)</span>, <i>Bull. Amer. Math. Soc.</i>, <b>66</b> (2): 94–97, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1960-10413-2">10.1090/s0002-9904-1960-10413-2</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Simultaneous+uniformization&rft.volume=66&rft.issue=2&rft.pages=94-97&rft.date=1960&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1960-10413-2&rft.aulast=Bers&rft.aufirst=Lipman&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fbulletin-of-the-american-mathematical-society%2Fvolume-66%2Fissue-2%2FSimultaneous-uniformization%2Fbams%2F1183523461.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBers1961" class="citation cs2">Bers, Lipman (1961), "Uniformization by Beltrami equations", <i>Comm. Pure Appl. Math.</i>, <b>14</b> (3): 215–228, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160140304">10.1002/cpa.3160140304</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comm.+Pure+Appl.+Math.&rft.atitle=Uniformization+by+Beltrami+equations&rft.volume=14&rft.issue=3&rft.pages=215-228&rft.date=1961&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160140304&rft.aulast=Bers&rft.aufirst=Lipman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBers1972" class="citation cs2"><a href="/wiki/Lipman_Bers" title="Lipman Bers">Bers, Lipman</a> (1972), "Uniformization, moduli, and Kleinian groups", <i>The Bulletin of the London Mathematical Society</i>, <b>4</b> (3): 257–300, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2F4.3.257">10.1112/blms/4.3.257</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-6093">0024-6093</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0348097">0348097</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Bulletin+of+the+London+Mathematical+Society&rft.atitle=Uniformization%2C+moduli%2C+and+Kleinian+groups&rft.volume=4&rft.issue=3&rft.pages=257-300&rft.date=1972&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0348097%23id-name%3DMR&rft.issn=0024-6093&rft_id=info%3Adoi%2F10.1112%2Fblms%2F4.3.257&rft.aulast=Bers&rft.aufirst=Lipman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Harmonic maps</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJost2006" class="citation cs2">Jost, Jürgen (2006), <i>Compact Riemann surfaces: an introduction to contemporary mathematics</i> (3rd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33065-3" title="Special:BookSources/978-3-540-33065-3"><bdi>978-3-540-33065-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Compact+Riemann+surfaces%3A+an+introduction+to+contemporary+mathematics&rft.edition=3rd&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-33065-3&rft.aulast=Jost&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Liouville's equation</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger1971" class="citation cs2">Berger, Melvyn S. (1971), "Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds", <i>Journal of Differential Geometry</i>, <b>5</b> (3–4): 325–332, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214429996">10.4310/jdg/1214429996</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Differential+Geometry&rft.atitle=Riemannian+structures+of+prescribed+Gaussian+curvature+for+compact+2-manifolds&rft.volume=5&rft.issue=3%E2%80%934&rft.pages=325-332&rft.date=1971&rft_id=info%3Adoi%2F10.4310%2Fjdg%2F1214429996&rft.aulast=Berger&rft.aufirst=Melvyn+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerger1977" class="citation cs2">Berger, Melvyn S. (1977), <i>Nonlinearity and functional analysis</i>, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-090350-4" title="Special:BookSources/978-0-12-090350-4"><bdi>978-0-12-090350-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinearity+and+functional+analysis&rft.pub=Academic+Press&rft.date=1977&rft.isbn=978-0-12-090350-4&rft.aulast=Berger&rft.aufirst=Melvyn+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor2011" class="citation cs2">Taylor, Michael E. (2011), <i>Partial differential equations III. Nonlinear equations</i>, Applied Mathematical Sciences, vol. 117 (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7048-0" title="Special:BookSources/978-1-4419-7048-0"><bdi>978-1-4419-7048-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations+III.+Nonlinear+equations&rft.series=Applied+Mathematical+Sciences&rft.edition=2nd&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-4419-7048-0&rft.aulast=Taylor&rft.aufirst=Michael+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> <p><b>Flows on Riemannian metrics</b> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1988" class="citation cs2">Hamilton, Richard S. (1988), "The Ricci flow on surfaces", <i>Mathematics and general relativity (Santa Cruz, CA, 1986)</i>, Contemp. Math., vol. 71, American Mathematical Society, pp. 237–262</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Ricci+flow+on+surfaces&rft.btitle=Mathematics+and+general+relativity+%28Santa+Cruz%2C+CA%2C+1986%29&rft.series=Contemp.+Math.&rft.pages=237-262&rft.pub=American+Mathematical+Society&rft.date=1988&rft.aulast=Hamilton&rft.aufirst=Richard+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChow1991" class="citation cs2">Chow, Bennett (1991), "The Ricci flow on the 2-sphere", <i>J. Differential Geom.</i>, <b>33</b> (2): 325–334, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214446319">10.4310/jdg/1214446319</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Differential+Geom.&rft.atitle=The+Ricci+flow+on+the+2-sphere&rft.volume=33&rft.issue=2&rft.pages=325-334&rft.date=1991&rft_id=info%3Adoi%2F10.4310%2Fjdg%2F1214446319&rft.aulast=Chow&rft.aufirst=Bennett&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsgoodPhillipsSarnak1988" class="citation cs2">Osgood, B.; Phillips, R.; Sarnak, P. (1988), "Extremals of determinants of Laplacians", <i>J. Funct. 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Soc.</i>, <b>6</b> (5): 771–782, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2032933">10.2307/2032933</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2032933">2032933</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Amer.+Math.+Soc.&rft.atitle=An+elementary+proof+of+the+existence+of+isothermal+parameters+on+a+surface&rft.volume=6&rft.issue=5&rft.pages=771-782&rft.date=1955&rft_id=info%3Adoi%2F10.2307%2F2032933&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2032933%23id-name%3DJSTOR&rft.aulast=Chern&rft.aufirst=Shiing-shen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeTurckKazdan1981" class="citation cs2">DeTurck, Dennis M.; <a href="/wiki/Jerry_Kazdan" title="Jerry Kazdan">Kazdan, Jerry L.</a> (1981), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASENS_1981_4_14_3_249_0">"Some regularity theorems in Riemannian geometry"</a>, <i>Annales Scientifiques de l'École Normale Supérieure</i>, Série 4, <b>14</b> (3): 249–260, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fasens.1405">10.24033/asens.1405</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-9593">0012-9593</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0644518">0644518</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=Some+regularity+theorems+in+Riemannian+geometry&rft.volume=14&rft.issue=3&rft.pages=249-260&rft.date=1981&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D644518%23id-name%3DMR&rft.issn=0012-9593&rft_id=info%3Adoi%2F10.24033%2Fasens.1405&rft.aulast=DeTurck&rft.aufirst=Dennis+M.&rft.au=Kazdan%2C+Jerry+L.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASENS_1981_4_14_3_249_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGusevskii2001" class="citation cs2">Gusevskii, N.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Uniformization">"Uniformization"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Uniformization&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Gusevskii&rft.aufirst=N.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DUniformization&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrushkalApanasovGusevskiĭ1986" class="citation cs2">Krushkal, S. L.; Apanasov, B. N.; Gusevskiĭ, N. A. (1986) [1981], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QvlhqAGN_y4C"><i>Kleinian groups and uniformization in examples and problems</i></a>, Translations of Mathematical Monographs, vol. 62, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4516-5" title="Special:BookSources/978-0-8218-4516-5"><bdi>978-0-8218-4516-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0647770">0647770</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kleinian+groups+and+uniformization+in+examples+and+problems&rft.place=Providence%2C+R.I.&rft.series=Translations+of+Mathematical+Monographs&rft.pub=American+Mathematical+Society&rft.date=1986&rft.isbn=978-0-8218-4516-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0647770%23id-name%3DMR&rft.aulast=Krushkal&rft.aufirst=S.+L.&rft.au=Apanasov%2C+B.+N.&rft.au=Gusevski%C4%AD%2C+N.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQvlhqAGN_y4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1996a" class="citation cs2"><a href="/wiki/Michael_E._Taylor" title="Michael E. Taylor">Taylor, Michael E.</a> (1996a), <i>Partial Differential Equations I: Basic Theory</i>, Springer, pp. 376–378, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94654-2" title="Special:BookSources/978-0-387-94654-2"><bdi>978-0-387-94654-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations+I%3A+Basic+Theory&rft.pages=376-378&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-94654-2&rft.aulast=Taylor&rft.aufirst=Michael+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1996b" class="citation cs2"><a href="/wiki/Michael_E._Taylor" title="Michael E. Taylor">Taylor, Michael E.</a> (1996b), <i>Partial Differential Equations II:Qualitative studies of linear equations</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94651-1" title="Special:BookSources/978-0-387-94651-1"><bdi>978-0-387-94651-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations+II%3AQualitative+studies+of+linear+equations&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-94651-1&rft.aulast=Taylor&rft.aufirst=Michael+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBersJohnSchechter1979" class="citation cs2">Bers, Lipman; John, Fritz; Schechter, Martin (1979), <i>Partial differential equations (reprint of the 1964 original)</i>, Lectures in Applied Mathematics, vol. 3A, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0049-2" title="Special:BookSources/978-0-8218-0049-2"><bdi>978-0-8218-0049-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations+%28reprint+of+the+1964+original%29&rft.series=Lectures+in+Applied+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1979&rft.isbn=978-0-8218-0049-2&rft.aulast=Bers&rft.aufirst=Lipman&rft.au=John%2C+Fritz&rft.au=Schechter%2C+Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffithsHarris1994" class="citation cs2">Griffiths, Phillip; Harris, Joseph (1994), <i>Principles of algebraic geometry</i>, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-05059-9" title="Special:BookSources/978-0-471-05059-9"><bdi>978-0-471-05059-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+algebraic+geometry&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-05059-9&rft.aulast=Griffiths&rft.aufirst=Phillip&rft.au=Harris%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner1983" class="citation cs2">Warner, Frank W. (1983), <i>Foundations of differentiable manifolds and Lie groups</i>, Graduate Texts in Mathematics, vol. 94, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-1799-0">10.1007/978-1-4757-1799-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90894-6" title="Special:BookSources/978-0-387-90894-6"><bdi>978-0-387-90894-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+differentiable+manifolds+and+Lie+groups&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1983&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-1799-0&rft.isbn=978-0-387-90894-6&rft.aulast=Warner&rft.aufirst=Frank+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniformization+theorem" class="Z3988"></span></li></ul> 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template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topological_manifold" title="Topological manifold">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Uniformization_theorem&oldid=1211249026">https://en.wikipedia.org/w/index.php?title=Uniformization_theorem&oldid=1211249026</a>"</div></div> <div id="catlinks" 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Template:Harvtxt"," 5.84% 40.025 2 Template:Pagetype"," 3.18% 21.745 2 Template:See_also"," 2.86% 19.553 21 Template:Math"," 2.74% 18.793 1 Template:Reflist"]},"scribunto":{"limitreport-timeusage":{"value":"0.499","limit":"10.000"},"limitreport-memusage":{"value":6782635,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAbikoff1981\"] = 1,\n [\"CITEREFAhlfors2006\"] = 1,\n [\"CITEREFAhlfors2010\"] = 1,\n [\"CITEREFAhlforsBers1960\"] = 1,\n [\"CITEREFAhlforsSario1960\"] = 1,\n [\"CITEREFAndrewsBryan2010\"] = 1,\n [\"CITEREFBeardon1984\"] = 1,\n [\"CITEREFBehnkeSommer1965\"] = 1,\n [\"CITEREFBerger1971\"] = 1,\n [\"CITEREFBerger1977\"] = 1,\n [\"CITEREFBers1960\"] = 1,\n [\"CITEREFBers1961\"] = 1,\n [\"CITEREFBers1972\"] = 1,\n [\"CITEREFBersJohnSchechter1979\"] = 1,\n [\"CITEREFBottazziniGray2013\"] = 1,\n [\"CITEREFBrendle2010\"] = 1,\n [\"CITEREFChang2000\"] = 1,\n [\"CITEREFChenLuTian2006\"] = 1,\n [\"CITEREFChern1955\"] = 1,\n [\"CITEREFChow1991\"] = 1,\n [\"CITEREFChrusciel1991\"] = 1,\n [\"CITEREFCourant1977\"] = 1,\n [\"CITEREFDeTurckKazdan1981\"] = 1,\n [\"CITEREFDonaldson2011\"] = 1,\n [\"CITEREFFarkasKra1980\"] = 1,\n [\"CITEREFForster1991\"] = 1,\n [\"CITEREFFreitag2011\"] = 1,\n [\"CITEREFGamelin2001\"] = 1,\n [\"CITEREFGray1994\"] = 1,\n [\"CITEREFGriffithsHarris1994\"] = 1,\n [\"CITEREFHamilton1988\"] = 1,\n [\"CITEREFHeins1949\"] = 1,\n [\"CITEREFHeins1951\"] = 1,\n [\"CITEREFHeins1957\"] = 1,\n [\"CITEREFHilbert1909\"] = 1,\n [\"CITEREFHubbard2006\"] = 1,\n [\"CITEREFJost2006\"] = 1,\n [\"CITEREFKlein1883\"] = 1,\n [\"CITEREFKodaira2007\"] = 1,\n [\"CITEREFKoebe1907a\"] = 1,\n [\"CITEREFKoebe1907b\"] = 1,\n [\"CITEREFKoebe1907c\"] = 1,\n [\"CITEREFKoebe1910a\"] = 1,\n [\"CITEREFKoebe1910b\"] = 1,\n [\"CITEREFKrushkalApanasovGusevskiĭ1986\"] = 1,\n [\"CITEREFMazzeoTaylor2002\"] = 1,\n [\"CITEREFNevanlinna1953\"] = 1,\n [\"CITEREFOsgoodPhillipsSarnak1988\"] = 1,\n [\"CITEREFPerron1923\"] = 1,\n [\"CITEREFPfluger1957\"] = 1,\n [\"CITEREFPoincaré1882\"] = 1,\n [\"CITEREFPoincaré1883\"] = 1,\n [\"CITEREFPoincaré1907\"] = 1,\n [\"CITEREFSario1952\"] = 1,\n [\"CITEREFSchlag2014\"] = 1,\n [\"CITEREFSchwarz1870\"] = 1,\n [\"CITEREFSiegel1988\"] = 1,\n [\"CITEREFSpringer1957\"] = 1,\n [\"CITEREFStruwe2002\"] = 1,\n [\"CITEREFTaylor1996a\"] = 1,\n [\"CITEREFTaylor1996b\"] = 1,\n [\"CITEREFTaylor2011\"] = 1,\n [\"CITEREFWarner1983\"] = 1,\n [\"CITEREFWeyl1913\"] = 1,\n [\"CITEREFWeyl1940\"] = 1,\n [\"CITEREFWeyl1964\"] = 1,\n [\"CITEREFde_Saint-Gervais2016\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Citation\"] = 67,\n [\"Eom\"] = 1,\n [\"Harvnb\"] = 3,\n [\"Harvs\"] = 5,\n [\"Harvtxt\"] = 12,\n [\"Manifolds\"] = 1,\n [\"Math\"] = 21,\n [\"Reflist\"] = 1,\n [\"See also\"] = 2,\n [\"Short description\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-2z5zb","timestamp":"20241124173829","ttl":2592000,"transientcontent":false}}});});</script> <script 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