Geometric Variational Problems and Scalar Curvature

所提出的研究的一个方面涉及具有标量曲率下界的流形的几何和拓扑。标量曲率是黎曼流形上最简单的曲率不变量。它表示黎曼流形中的小测地球的体积与欧几里德空间中的标准球的体积偏离的量。标量曲率也存在于自然科学中。例如，在广义相对论中，它是爱因斯坦-希尔伯特作用量的拉格朗日密度。几何、拓扑学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。另一个主要的研究领域涉及极小曲面。极小曲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，极小表面作为黑洞的“视界”出现；肥皂膜和毛细管界面也提供了极小表面的例子。PI将研究极小曲面的存在性、正则性和拓扑学。这两个方面是通过几何变分理论紧密联系在一起的。项目涉及一系列关于微分几何、几何测度论和偏微分方程的主题。几何研究的一个主要主题是利用黎曼多面体建立标量曲率的几何比较定理，目的是在低正则空间上定义正标量曲率的弱概念。PI计划继续研究更一般的多面体，特别是高维单形的这样一个定理，以及它与广义相对论中准局部质量的联系。PI还计划继续他对具有正标量曲率和平均凸边界的流形的模空间结构的研究，包括研究它的高同伦群，以及由其他相关曲率条件定义的模空间的结构。此外，PI将研究具有标量曲率下界的奇异空间，并将这种奇异流形理解为具有相同假设的光滑流形的特定极限。PI研究的一个核心工具是最小方差理论。PI计划了解在一般的Lipschitz域中，特别是在局部凸多面体区域中，具有自由边界和毛细边界条件的极小曲面的存在性和正则性。他还计划通过最小-最大构造建立毛细表面的一般存在理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.