Document Zbl 1544.53053

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Am. Math. Soc.</a> <a href="/?q=in%3A514801" title="Articles in this Issue">377, No. 3, 1989-2007 (2024)</a>. </div> <div class="abstract">An initial data set $(M,g,K)$ consists of a smooth orientable $n$-dimensional manifold $M$ equipped with a Riemannian metric $g$ and a symmetric $(0,2)$-tensor $K$. The main physical example is when $(M,g,K)$ is an initial data set in a spacetime, i.e., $M$ is a space-like hypersurface in a Lorentzian manifold $\overline{M}$, with induced metric $g$ and second fundamental form $K$. The local energy density $\mu$ and the local current density $J$ of an initial data set $(M,g,K)$ are given by \[ \mu =12(S-|K|^2+(\mathrm{tr}K)^2) \ \ \ \text{and} \ \ \ J=\mathrm{div}(K-\mathrm{tr}(K)g) \ , \] where $S$ is the scalar curvature of $(M,g)$. When $(M,g,K)$ is a spacetime initial data set, these quantities are given by $\mu=G(u,u)$, $J=G(u,\cdot)$, where $G$ is the Einstein tensor.<br class="zbmathjax-paragraph">Let $\Sigma$ be a closed two-sided surface in $M$. We say that $\Sigma$ is an outer trapped surface if $\theta_+=\mathrm{tr}_{\Sigma}(K)+H < 0$. If $\theta_+$ vanishes identically we say that $\Sigma$ is a marginally outer trapped surface (or MOTS). <br class="zbmathjax-paragraph">The paper under review presents some initial data rigidity results for compact initial data sets. For example, in the main theorem the authors prove that if $(M, g, K)$ satisfies the energy condition $\mu - |J|\geq c$ for some constant $c > 0$ and the boundary of $M$ can be expressed as a disjoint union $\partial M = \Sigma_0 \cup S$ of nonempty unions of components such that the following conditions hold: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">1.</dt><dd class="zbmathjax-item1"> $\theta_+\leq 0$ on $\Sigma_0$ with respect to the normal vector that points into $M$;</dd><dt class="zbmathjax-item1">2.</dt><dd class="zbmathjax-item1"> $\theta_+\geq 0$ on $S$ with respect to the normal vector that points out of $M$;</dd><dt class="zbmathjax-item1">3.</dt><dd class="zbmathjax-item1"> There exists a continuous map $\rho : M \to\Sigma$ such that $\rho \circ i: \Sigma\to \Sigma$ is homotopic to $\mathrm{id}_{\Sigma}$, where $i :\Sigma \hookrightarrow M$ is the inclusion map;</dd><dt class="zbmathjax-item1">4.</dt><dd class="zbmathjax-item1"> The relative homology group $H^2(M, \Sigma_0)$ vanishes;</dd><dt class="zbmathjax-item1">5.</dt><dd class="zbmathjax-item1"> $\Sigma_0$ minimizes area.</dd></dl> Then $\Sigma_0$ is topologically a sphere and its area satisfies $A(\Sigma_0) \leq 4\pi/c$. Moreover, if $A(\Sigma_0) = 4\pi/c$, then: <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1"> $(M, g)$ is isometric to $([0,\ell] \times \Sigma_0, dt^2 + g_0)$ for some $\ell > 0$, where $g_0$ is the induced metric on $\Sigma_0$ and has constant Gaussian curvature $c$;</dd><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1"> $K = a dt^2$ on $M$, where $a \in C^{\infty}(M)$ that depends only on $t\in [0, \ell]$;</dd><dt class="zbmathjax-item1">●</dt><dd class="zbmathjax-item1"> $\mu = c$ and $J = 0$ on $M$.</dd></dl> The proof uses in a crucial way the theory of MOTS in initial data sets.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A19416">Andrea Tamburelli (Houston)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7876010">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A53C24" title="MSC2020">53C24</a> </td> <td class="space"> Rigidity results </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amarginally+outer+trapped+surfaces">marginally outer trapped surfaces</a>; <a href="/?q=ut%3Ainitial+data+sets">initial data sets</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1544.53053" data-ciurl="/ci/07876010" data-biburl="/bibtex/07876010.bib" data-amsurl="/amsrefs/07876010.bib" data-xmlurl="/xml/07876010.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07876010.pdf" title="Zbl 1544.53053 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1090/tran/9079" aria-label="DOI for “Some rigidity results for compact initial data sets”" title="10.1090/tran/9079">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2301.00639"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Andersson, Lars, Complex analysis and dynamical systems IV. 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