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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Ye, R"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.05778">arXiv:2408.05778</a> <span> [<a href="https://arxiv.org/pdf/2408.05778">pdf</a>, <a href="https://arxiv.org/format/2408.05778">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Pareto Front Shape-Agnostic Pareto Set Learning in Multi-Objective Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongguang Ye</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Longcan Chen</a>, <a href="/search/math?searchtype=author&query=Kou%2C+W">Wei-Bin Kou</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinyuan Zhang</a>, <a href="/search/math?searchtype=author&query=Ishibuchi%2C+H">Hisao Ishibuchi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.05778v1-abstract-short" style="display: inline;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high pe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05778v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05778v1-abstract-full" style="display: none;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'none'; document.getElementById('2408.05778v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IEEE International Conference on Systems, Man, and Cybernetics (IEEE SMC 2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.01224">arXiv:2404.01224</a> <span> [<a href="https://arxiv.org/pdf/2404.01224">pdf</a>, <a href="https://arxiv.org/format/2404.01224">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Collaborative Pareto Set Learning in Multiple Multi-Objective Optimization Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shang%2C+C">Chikai Shang</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongguang Ye</a>, <a href="/search/math?searchtype=author&query=Jiang%2C+J">Jiaqi Jiang</a>, <a href="/search/math?searchtype=author&query=Gu%2C+F">Fangqing Gu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.01224v2-abstract-short" style="display: inline;"> Pareto Set Learning (PSL) is an emerging research area in multi-objective optimization, focusing on training neural networks to learn the mapping from preference vectors to Pareto optimal solutions. However, existing PSL methods are limited to addressing a single Multi-objective Optimization Problem (MOP) at a time. When faced with multiple MOPs, this limitation results in significant inefficienci… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.01224v2-abstract-full').style.display = 'inline'; document.getElementById('2404.01224v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.01224v2-abstract-full" style="display: none;"> Pareto Set Learning (PSL) is an emerging research area in multi-objective optimization, focusing on training neural networks to learn the mapping from preference vectors to Pareto optimal solutions. However, existing PSL methods are limited to addressing a single Multi-objective Optimization Problem (MOP) at a time. When faced with multiple MOPs, this limitation results in significant inefficiencies and hinders the ability to exploit potential synergies across varying MOPs. In this paper, we propose a Collaborative Pareto Set Learning (CoPSL) framework, which learns the Pareto sets of multiple MOPs simultaneously in a collaborative manner. CoPSL particularly employs an architecture consisting of shared and MOP-specific layers. The shared layers are designed to capture commonalities among MOPs collaboratively, while the MOP-specific layers tailor these general insights to generate solution sets for individual MOPs. This collaborative approach enables CoPSL to efficiently learn the Pareto sets of multiple MOPs in a single execution while leveraging the potential relationships among various MOPs. To further understand these relationships, we experimentally demonstrate that shareable representations exist among MOPs. Leveraging these shared representations effectively improves the capability to approximate Pareto sets. Extensive experiments underscore the superior efficiency and robustness of CoPSL in approximating Pareto sets compared to state-of-the-art approaches on a variety of synthetic and real-world MOPs. Code is available at https://github.com/ckshang/CoPSL. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.01224v2-abstract-full').style.display = 'none'; document.getElementById('2404.01224v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted by IJCNN 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.05107">arXiv:2203.05107</a> <span> [<a href="https://arxiv.org/pdf/2203.05107">pdf</a>, <a href="https://arxiv.org/ps/2203.05107">ps</a>, <a href="https://arxiv.org/format/2203.05107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Ricci Flow and Gromov Almost Flat Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+E">Eric Chen</a>, <a href="/search/math?searchtype=author&query=Wei%2C+G">Guofang Wei</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.05107v1-abstract-short" style="display: inline;"> We employ the Ricci flow to derive a new theorem about Gromov almost flat manifolds, which generalizes and strengthens the celebrated Gromov--Ruh Theorem. In our theorem, the condition $diam^2 |K| \leq 蔚_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $\|Rm\|_{n/2}$ $ C_S^2 \leq \varepsilon_n$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.05107v1-abstract-full" style="display: none;"> We employ the Ricci flow to derive a new theorem about Gromov almost flat manifolds, which generalizes and strengthens the celebrated Gromov--Ruh Theorem. In our theorem, the condition $diam^2 |K| \leq 蔚_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $\|Rm\|_{n/2}$ $ C_S^2 \leq \varepsilon_n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.05107v1-abstract-full').style.display = 'none'; document.getElementById('2203.05107v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53E20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.10482">arXiv:1908.10482</a> <span> [<a href="https://arxiv.org/pdf/1908.10482">pdf</a>, <a href="https://arxiv.org/ps/1908.10482">ps</a>, <a href="https://arxiv.org/format/1908.10482">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11139-021-00476-x">10.1007/s11139-021-00476-x <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongqing Ye</a>, <a href="/search/math?searchtype=author&query=Zelingher%2C+E">Elad Zelingher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1908.10482v2-abstract-short" style="display: inline;"> We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.10482v2-abstract-full" style="display: none;"> We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.10482v2-abstract-full').style.display = 'none'; document.getElementById('1908.10482v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages. Comments are welcome. v2: This version contains all changes that were made for the Ramanujan Journal submission</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E50; 11F66 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.10413">arXiv:1903.10413</a> <span> [<a href="https://arxiv.org/pdf/1903.10413">pdf</a>, <a href="https://arxiv.org/ps/1903.10413">ps</a>, <a href="https://arxiv.org/format/1903.10413">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jnt.2020.06.007">10.1016/j.jnt.2020.06.007 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Epsilon factors of representations of finite general linear groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongqing Ye</a>, <a href="/search/math?searchtype=author&query=Zelingher%2C+E">Elad Zelingher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.10413v2-abstract-short" style="display: inline;"> We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.10413v2-abstract-full').style.display = 'inline'; document.getElementById('1903.10413v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.10413v2-abstract-full" style="display: none;"> We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We prove that these exterior square factors coincide in a special case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.10413v2-abstract-full').style.display = 'none'; document.getElementById('1903.10413v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages. Comments are welcome. Changes from v1: This version contains all changes that were made for the Journal of Number Theory submission</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11L05; 20C33 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1807.04816">arXiv:1807.04816</a> <span> [<a href="https://arxiv.org/pdf/1807.04816">pdf</a>, <a href="https://arxiv.org/ps/1807.04816">ps</a>, <a href="https://arxiv.org/format/1807.04816">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11856-020-2084-y">10.1007/s11856-020-2084-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: finite field analogs and level zero representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongqing Ye</a>, <a href="/search/math?searchtype=author&query=Zelingher%2C+E">Elad Zelingher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1807.04816v2-abstract-short" style="display: inline;"> We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.04816v2-abstract-full').style.display = 'inline'; document.getElementById('1807.04816v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1807.04816v2-abstract-full" style="display: none;"> We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.04816v2-abstract-full').style.display = 'none'; document.getElementById('1807.04816v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">42 pages. Comments are welcome. Changes from v1: This version contains all changes that were made for the Israel Journal of Mathematics submission. The main difference between the arXiv version and the Israel Journal of Mathematics version is that the arXiv version contains proofs and details for the odd case, and also contains Section 2.5</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20C33; 22E50; 11F66 (Primary); 11L05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.01664">arXiv:1804.01664</a> <span> [<a href="https://arxiv.org/pdf/1804.01664">pdf</a>, <a href="https://arxiv.org/ps/1804.01664">ps</a>, <a href="https://arxiv.org/format/1804.01664">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Rankin-Selberg gamma factors of level zero representations of $GL_n$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongqing Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.01664v1-abstract-short" style="display: inline;"> For a $p$-adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$, we prove that the Rankin-Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$. Our results can be used to prove a variant of Jacquet's conjecture. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.01664v1-abstract-full" style="display: none;"> For a $p$-adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$, we prove that the Rankin-Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$. Our results can be used to prove a variant of Jacquet's conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.01664v1-abstract-full').style.display = 'none'; document.getElementById('1804.01664v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.02074">arXiv:1601.02074</a> <span>  </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the Very-well-poised Bilateral Basic Hypergeometric $_5蠄_5$ Series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Runping Ye</a>, <a href="/search/math?searchtype=author&query=Zou%2C+Q">Qing Zou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1601.02074v2-abstract-short" style="display: inline;"> In this paper, we prove several transformation formulas for the very-well-poised bilateral basic hypergeometric $_5蠄_5$ series by using the relationship between the bilateral basic hypergeometric $_5蠄_5$ series and basic hypergeometric $_8蠁_7$ series. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.02074v2-abstract-full" style="display: none;"> In this paper, we prove several transformation formulas for the very-well-poised bilateral basic hypergeometric $_5蠄_5$ series by using the relationship between the bilateral basic hypergeometric $_5蠄_5$ series and basic hypergeometric $_8蠁_7$ series. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.02074v2-abstract-full').style.display = 'none'; document.getElementById('1601.02074v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This paper has been withdrawn by the author due to some errors in the Theorems</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1203.5307">arXiv:1203.5307</a> <span> [<a href="https://arxiv.org/pdf/1203.5307">pdf</a>, <a href="https://arxiv.org/ps/1203.5307">ps</a>, <a href="https://arxiv.org/format/1203.5307">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> A Note On Obata's Rigidity Theorem I </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+G">Guoqiang Wu</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1203.5307v2-abstract-short" style="display: inline;"> In this note we present various extensions of Obata's rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity case we also characterize the geometry and topology of the underlying manifold in more general sit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.5307v2-abstract-full').style.display = 'inline'; document.getElementById('1203.5307v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1203.5307v2-abstract-full" style="display: none;"> In this note we present various extensions of Obata's rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity case we also characterize the geometry and topology of the underlying manifold in more general situations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1203.5307v2-abstract-full').style.display = 'none'; document.getElementById('1203.5307v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 April, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 March, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Some remarks are added. A few typos are removed. The proof of one lemma is modified</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0912.0074">arXiv:0912.0074</a> <span> [<a href="https://arxiv.org/pdf/0912.0074">pdf</a>, <a href="https://arxiv.org/ps/0912.0074">ps</a>, <a href="https://arxiv.org/format/0912.0074">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Existence, Convergence and Limit Map of the Laplacian Flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+F">Feng Xu</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0912.0074v1-abstract-short" style="display: inline;"> We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed $G_2$-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free $G_2$-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsion-free $G_2$-structures. We also present a number of results which constitute… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0912.0074v1-abstract-full').style.display = 'inline'; document.getElementById('0912.0074v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0912.0074v1-abstract-full" style="display: none;"> We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed $G_2$-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free $G_2$-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsion-free $G_2$-structures. We also present a number of results which constitute a fairly complete algebraic and analytic basis for studying the Laplacian flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0912.0074v1-abstract-full').style.display = 'none'; document.getElementById('0912.0074v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 December, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">41 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0709.2724">arXiv:0709.2724</a> <span> [<a href="https://arxiv.org/pdf/0709.2724">pdf</a>, <a href="https://arxiv.org/ps/0709.2724">ps</a>, <a href="https://arxiv.org/format/0709.2724">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0709.2724v1-abstract-short" style="display: inline;"> In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0709.2724v1-abstract-full" style="display: none;"> In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0709.2724v1-abstract-full').style.display = 'none'; document.getElementById('0709.2724v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 September, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This survey was written for the 4th International Congress of Chinese Mathematicians to be held in Hangzhou, China in December 2007, and for the memorial volume dedicated to the 771 class of mathematics of China University of Science and Technology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0709.0512">arXiv:0709.0512</a> <span> [<a href="https://arxiv.org/pdf/0709.0512">pdf</a>, <a href="https://arxiv.org/ps/0709.0512">ps</a>, <a href="https://arxiv.org/format/0709.0512">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Sobolev Inequalities, Riesz Transforms and the Ricci Flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0709.0512v1-abstract-short" style="display: inline;"> In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow to $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general p. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0709.0512v1-abstract-full" style="display: none;"> In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow to $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general p. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0709.0512v1-abstract-full').style.display = 'none'; document.getElementById('0709.0512v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 September, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0708.2008">arXiv:0708.2008</a> <span> [<a href="https://arxiv.org/pdf/0708.2008">pdf</a>, <a href="https://arxiv.org/ps/0708.2008">ps</a>, <a href="https://arxiv.org/format/0708.2008">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The Log Entropy Functional Along the Ricci Flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0708.2008v3-abstract-short" style="display: inline;"> In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0708.2008v3-abstract-full" style="display: none;"> In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0708.2008v3-abstract-full').style.display = 'none'; document.getElementById('0708.2008v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 August, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">A typo in the monotonicity inequality for the log entropy functional is corrected. Namely the factor 1/(16 蠅) should be (4 蠅)/n. (From the given computations it is obvious that this number should be (4蠅)/n.)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0708.2005">arXiv:0708.2005</a> <span> [<a href="https://arxiv.org/pdf/0708.2005">pdf</a>, <a href="https://arxiv.org/ps/0708.2005">ps</a>, <a href="https://arxiv.org/format/0708.2005">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The logarithmic Sobolev inequality along the Ricci flow: the case $位_0(g_0)=0$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0708.2005v1-abstract-short" style="display: inline;"> We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $位_0(g_0)>0$ to the case $位_0(g_0)=0$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0708.2005v1-abstract-full" style="display: none;"> We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $位_0(g_0)>0$ to the case $位_0(g_0)=0$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0708.2005v1-abstract-full').style.display = 'none'; document.getElementById('0708.2005v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0708.2003">arXiv:0708.2003</a> <span> [<a href="https://arxiv.org/pdf/0708.2003">pdf</a>, <a href="https://arxiv.org/ps/0708.2003">ps</a>, <a href="https://arxiv.org/format/0708.2003">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The logarithmic Sobolev inequality along the Ricci flow in dimension 2 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0708.2003v1-abstract-short" style="display: inline;"> In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0708.2003v1-abstract-full" style="display: none;"> In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0708.2003v1-abstract-full').style.display = 'none'; document.getElementById('0708.2003v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0707.2424">arXiv:0707.2424</a> <span> [<a href="https://arxiv.org/pdf/0707.2424">pdf</a>, <a href="https://arxiv.org/ps/0707.2424">ps</a>, <a href="https://arxiv.org/format/0707.2424">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The logarithmic Sobolev inequality along the Ricci flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0707.2424v4-abstract-short" style="display: inline;"> We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing esti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.2424v4-abstract-full').style.display = 'inline'; document.getElementById('0707.2424v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0707.2424v4-abstract-full" style="display: none;"> We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.2424v4-abstract-full').style.display = 'none'; document.getElementById('0707.2424v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 August, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 July, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">One appendix is added. A theorem on the Sobolev inequality along the Ricci flow with surgeries of Perelman is added. Two nonlocal Sobolev inequalities are also added. These results were obtained at the time of the posting of the first version of the paper. They were originally planned as parts of two upcoming papers of the author</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20 (Primary); 53C21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0703505">arXiv:math/0703505</a> <span> [<a href="https://arxiv.org/pdf/math/0703505">pdf</a>, <a href="https://arxiv.org/ps/math/0703505">ps</a>, <a href="https://arxiv.org/format/math/0703505">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+G">Guofang Wei</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0703505v2-abstract-short" style="display: inline;"> In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curvature bounded from below and diameter bounded from above to yield a maximum es… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0703505v2-abstract-full').style.display = 'inline'; document.getElementById('math/0703505v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0703505v2-abstract-full" style="display: none;"> In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curvature bounded from below and diameter bounded from above to yield a maximum estimate without dependence on a positive lower bound for the volume. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0703505v2-abstract-full').style.display = 'none'; document.getElementById('math/0703505v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 November, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 March, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2007. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">In Theorem A, the previous maximum estimate in terms of the isoperimetric constant is replaced by a maximum estimate in terms of the volume-normalized isoperimetric constant. The statements of Gallot's estimate for the isoperimetric constant are corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53CXX </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0609321">arXiv:math/0609321</a> <span> [<a href="https://arxiv.org/pdf/math/0609321">pdf</a>, <a href="https://arxiv.org/ps/math/0609321">ps</a>, <a href="https://arxiv.org/format/math/0609321">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> On the l-Function and the Reduced volume of Perelman II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0609321v2-abstract-short" style="display: inline;"> In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0609321v2-abstract-full" style="display: none;"> In this paper we present a major application of the l-function and the reduced volume of Perelman, namely their application to the analysis of the asymptotical limits of kappa solutions of the Ricci flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0609321v2-abstract-full').style.display = 'none'; document.getElementById('math/0609321v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2006; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2006. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This paper has been available at the author's website. We post it here to make it easier to access. To appear in Transactions of American Mathematical Society</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0609320">arXiv:math/0609320</a> <span> [<a href="https://arxiv.org/pdf/math/0609320">pdf</a>, <a href="https://arxiv.org/ps/math/0609320">ps</a>, <a href="https://arxiv.org/format/math/0609320">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> On the l-Function and the Reduced Volume of Perelman I </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0609320v2-abstract-short" style="display: inline;"> The main purpose of this paper is to present a number of analytic and geometric properties of the $l$-function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the reduced volume. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0609320v2-abstract-full" style="display: none;"> The main purpose of this paper is to present a number of analytic and geometric properties of the $l$-function and the reduced volume of Perelman, including in particular the monotonicity, the upper bound and the rigidities of the reduced volume. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0609320v2-abstract-full').style.display = 'none'; document.getElementById('math/0609320v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2006; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2006. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This paper has been available at the author's website. We post it here to make it easier to access. To appear in Transactions of American Mathematical Society</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0509143">arXiv:math/0509143</a> <span> [<a href="https://arxiv.org/pdf/math/0509143">pdf</a>, <a href="https://arxiv.org/ps/math/0509143">ps</a>, <a href="https://arxiv.org/format/math/0509143">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Curvature Estimates for the Ricci Flow II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0509143v4-abstract-short" style="display: inline;"> In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the norm of the Riemann curvature tensor. They also serve as characterizations of blo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0509143v4-abstract-full').style.display = 'inline'; document.getElementById('math/0509143v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0509143v4-abstract-full" style="display: none;"> In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the norm of the Riemann curvature tensor. They also serve as characterizations of blow-up singularities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0509143v4-abstract-full').style.display = 'none'; document.getElementById('math/0509143v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 July, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 September, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20 (Primary); 53C21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0509142">arXiv:math/0509142</a> <span> [<a href="https://arxiv.org/pdf/math/0509142">pdf</a>, <a href="https://arxiv.org/ps/math/0509142">ps</a>, <a href="https://arxiv.org/format/math/0509142">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Curvature Estimates for the Ricci Flow I </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0509142v4-abstract-short" style="display: inline;"> In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0509142v4-abstract-full" style="display: none;"> In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0509142v4-abstract-full').style.display = 'none'; document.getElementById('math/0509142v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 July, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 September, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2005. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20 (Primary); 52C21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/9901059">arXiv:math/9901059</a> <span> [<a href="https://arxiv.org/pdf/math/9901059">pdf</a>, <a href="https://arxiv.org/ps/math/9901059">ps</a>, <a href="https://arxiv.org/format/math/9901059">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Equivariant and Bott-type Seiberg-Witten Floer Homology: Part II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/9901059v2-abstract-short" style="display: inline;"> We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/9901059v2-abstract-full').style.display = 'inline'; document.getElementById('math/9901059v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/9901059v2-abstract-full" style="display: none;"> We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/9901059v2-abstract-full').style.display = 'none'; document.getElementById('math/9901059v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 1999; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 January, 1999; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 1999. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">41 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/9901058">arXiv:math/9901058</a> <span> [<a href="https://arxiv.org/pdf/math/9901058">pdf</a>, <a href="https://arxiv.org/ps/math/9901058">ps</a>, <a href="https://arxiv.org/format/math/9901058">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Equivariant and Bott-type Seiberg-Witten Floer Homology: Part I </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+G">Guofang Wang</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/9901058v2-abstract-short" style="display: inline;"> We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/9901058v2-abstract-full" style="display: none;"> We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/9901058v2-abstract-full').style.display = 'none'; document.getElementById('math/9901058v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 1999; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 January, 1999; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 1999. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">AMS Tex, 49 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/dg-ga/9709020">arXiv:dg-ga/9709020</a> <span> [<a href="https://arxiv.org/pdf/dg-ga/9709020">pdf</a>, <a href="https://arxiv.org/ps/dg-ga/9709020">ps</a>, <a href="https://arxiv.org/format/dg-ga/9709020">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Foliation by Constant Mean Curvature Spheres on Asymptotically Flat Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="dg-ga/9709020v1-abstract-short" style="display: inline;"> In this paper, the existence and uniqueness of foliations by constant mean curvature spheres on asymptotically flat manifolds of nonzero ADM mass in all dimensions were established. (A similar result in the case of positive mass was obtained independently by G. Huisken and S. T. Yau, see the introduction of this paper and their paper in Inv. Math.) </span> <span class="abstract-full has-text-grey-dark mathjax" id="dg-ga/9709020v1-abstract-full" style="display: none;"> In this paper, the existence and uniqueness of foliations by constant mean curvature spheres on asymptotically flat manifolds of nonzero ADM mass in all dimensions were established. (A similar result in the case of positive mass was obtained independently by G. Huisken and S. T. Yau, see the introduction of this paper and their paper in Inv. Math.) <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9709020v1-abstract-full').style.display = 'none'; document.getElementById('dg-ga/9709020v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 1997; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 1997. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">AMS-Tex. This paper has appeared in the volume "Geometric Analysis and Calculus of Variation" (dedicated to S. Hildebrandt on the occation of his 60th birthday, edit. J. Jost, Springer, 1996). For the convenience of readers, we post the paper here</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/dg-ga/9708014">arXiv:dg-ga/9708014</a> <span> [<a href="https://arxiv.org/pdf/dg-ga/9708014">pdf</a>, <a href="https://arxiv.org/ps/dg-ga/9708014">ps</a>, <a href="https://arxiv.org/format/dg-ga/9708014">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> On the geometry and topology of manifolds of positive bi-Ricci curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shen%2C+Y">Ying Shen</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="dg-ga/9708014v1-abstract-short" style="display: inline;"> We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some topological applications are discussed. We also give examples of manifolds of pos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9708014v1-abstract-full').style.display = 'inline'; document.getElementById('dg-ga/9708014v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="dg-ga/9708014v1-abstract-full" style="display: none;"> We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some topological applications are discussed. We also give examples of manifolds of positive bi-Ricci curvature and prove that the connect sum of manifolds of positive bi-Ricci curvature admits metrics of positive bi-Ricci curvature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9708014v1-abstract-full').style.display = 'none'; document.getElementById('dg-ga/9708014v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 August, 1997; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 1997. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C21; 53C42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/dg-ga/9701010">arXiv:dg-ga/9701010</a> <span> [<a href="https://arxiv.org/pdf/dg-ga/9701010">pdf</a>, <a href="https://arxiv.org/ps/dg-ga/9701010">ps</a>, <a href="https://arxiv.org/format/dg-ga/9701010">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Bott-type and equivariant Seiberg-Witten Floer homology I </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+G">G. Wang</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">R. Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="dg-ga/9701010v2-abstract-short" style="display: inline;"> We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance. </span> <span class="abstract-full has-text-grey-dark mathjax" id="dg-ga/9701010v2-abstract-full" style="display: none;"> We construct Bott-type and stable equivariant Seiberg-Witten Floer homology and cohomology for rational homology spheres, and prove their diffeomorphism invariance. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9701010v2-abstract-full').style.display = 'none'; document.getElementById('dg-ga/9701010v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 January, 1997; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 January, 1997; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 1997. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">AMS Tex, 45 pages. A few misprints were removed from the last version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/dg-ga/9508012">arXiv:dg-ga/9508012</a> <span> [<a href="https://arxiv.org/pdf/dg-ga/9508012">pdf</a>, <a href="https://arxiv.org/ps/dg-ga/9508012">ps</a>, <a href="https://arxiv.org/format/dg-ga/9508012">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Controlled Geometry via Smoothing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Petersen%2C+P">Peter Petersen</a>, <a href="/search/math?searchtype=author&query=Wei%2C+G">Guofang Wei</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="dg-ga/9508012v1-abstract-short" style="display: inline;"> We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained. </span> <span class="abstract-full has-text-grey-dark mathjax" id="dg-ga/9508012v1-abstract-full" style="display: none;"> We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9508012v1-abstract-full').style.display = 'none'; document.getElementById('dg-ga/9508012v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 August, 1995; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 1995. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, Latex</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/dg-ga/9411014">arXiv:dg-ga/9411014</a> <span> [<a href="https://arxiv.org/pdf/dg-ga/9411014">pdf</a>, <a href="https://arxiv.org/ps/dg-ga/9411014">ps</a>, <a href="https://arxiv.org/format/dg-ga/9411014">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Smoothing Riemannian Metrics with Ricci Curvature Bounds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dai%2C+X">Xianzhe Dai</a>, <a href="/search/math?searchtype=author&query=Wei%2C+G">Guofang Wei</a>, <a href="/search/math?searchtype=author&query=Ye%2C+R">Rugang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="dg-ga/9411014v1-abstract-short" style="display: inline;"> We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds. </span> <span class="abstract-full has-text-grey-dark mathjax" id="dg-ga/9411014v1-abstract-full" style="display: none;"> We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('dg-ga/9411014v1-abstract-full').style.display = 'none'; document.getElementById('dg-ga/9411014v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 1994; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 1994. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, latex</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns 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