Geometric Variational Problems and Scalar Curvature

几何变分问题和标量曲率

• 批准号: 2202343
• 负责人: Chao Li
• 金额: $ 17.45万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-11-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2202343&HistoricalAwards=false
• 关键词: Geometric; Variational; Problems; Scalar; Curvature

One aspect of the proposed research has to do with the geometry and topology of manifolds with scalar curvature lower bounds. Scalar curvature is the simplest curvature invariant of a Riemannian manifold. It represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. Scalar curvature also arises in natural sciences. For instance, in general relativity, it is the Lagrangian density of the Einstein-Hilbert action. A natural and deep question in geometry, topology and mathematical physics is to understand the affect of scalar curvature conditions on a manifold. The other main area of investigation concerns minimal surfaces. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In mathematical model of general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. The PI will investigate the existence, regularity and topology of minimal surfaces. The two aspects proposed here are deeply connected via geometric variational theory.The project concerns a range topics on differential geometry, geometric measure theory and partial differential equations. A main theme of the research in geometry will be a geometric comparison theorem for scalar curvature using Riemannian polyhedra, with the aim to define weak notions of positive scalar curvature on spaces with low regularity. The PI plans to continue his investigations into such a theorem for more general polytopes, especially simplexes of higher dimensions, and its connection to quasi-local mass in general relativity. The PI also plans to continue his investigation on the structure of moduli spaces of manifolds with positive scalar curvature and mean convex boundary, including studying its high homotopy groups, and the structure of moduli spaces defined by other related curvature conditions. In addition, the PI will study singular spaces with scalar curvature lower bounds, and understand when such a singular manifold arises as a certain limit of smooth manifolds with same assumptions. A central tool in the PI’s research is the theory of minimal varieties. The PI plans to understand the existence and regularity of minimal surfaces with free boundary and capillary boundary conditions in general Lipschitz domains, especially in locally convex polyhedral domains. He also plans to establish a general existence theory of capillary surfaces via a min-max construction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

所提出的研究的一个方面与具有标量曲率下界的流形的几何和拓扑有关。标量曲率是黎曼流形的最简单的曲率不变量，它表示黎曼流形中小测地球的体积。流形与欧几里德空间中标准球的流形的偏差也出现在自然科学中。例如，在广义相对论中，爱因斯坦-希尔伯特作用的拉格朗日密度是几何、拓扑和数学物理中一个自然而深刻的问题，是理解标量曲率条件对流形的影响。研究涉及极小表面。极小表面是自然界中许多界面的数学模型，在广义相对论的数学模型中，极小表面表现为黑洞的“视界”和毛细管界面。还提供了最小曲面的示例。PI 将研究最小曲面的存在性、规律性和拓扑结构，这里提出的两个方面通过几何变分理论紧密相连。该项目涉及微分几何、几何测度理论和偏微分等一系列主题。几何研究的一个主题是使用黎曼多面体的标量曲率的几何比较定理，旨在定义低正则空间上的弱概念 PI 平面。继续研究更一般的多面体定理，特别是更高维度的单纯形，及其与广义相对论中准局域质量的联系。 PI 还计划继续研究正标量流形的模空间结构。曲率和平均凸边界，包括研究其高同伦群，以及其他相关曲率条件定义的模空间的结构。此外，PI还将研究具有标量的奇异空间。 PI 的研究的核心工具是最小簇理论，PI 计划了解最小曲面的存在性和规律性。一般Lipschitz域中的自由边界和毛细管边界条件，特别是局部凸多面体域中的自由边界和毛细管边界条件，他还计划通过最小-最大构造建立毛细管表面的一般存在理论。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。

项目成果

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

双曲空间中抛物面多面体的二面刚度

• DOI: 10.3842/sigma.2020.099
• 发表时间: 2020-10
• 期刊: Integrability and Geometry: Methods and Applications
• 作者: Li; Chao
• 通讯作者: Chao

Classifying sufficiently connected PSC manifoldsin 4 and 5 dimensions

对 4 维和 5 维中充分连接的 PSC 流形进行分类

• DOI: 10.2140/gt.2023.27.1635
• 发表时间: 2023-01
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Chodosh, Otis;Li, Chao;Liokumovich, Yevgeny
• 通讯作者: Liokumovich, Yevgeny

Stable anisotropic minimal hypersurfaces in

稳定各向异性最小超曲面

• DOI: 10.1017/fmp.2023.1
• 发表时间: 2023-01
• 期刊: Pi
• 作者: Chodosh, Otis;Li, Chao
• 通讯作者: Li, Chao

Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains

局部多面体域中自由边界极小曲面的正则性

• DOI: 10.1002/cpa.22039
• 发表时间: 2022-05
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Edelen, Nick;Li, Chao
• 通讯作者: Li, Chao