Scalar curvature and geometric variational problems

标量曲率和几何变分问题

• 批准号: 2303624
• 负责人: Chao Li
• 金额: $ 38.05万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303624&HistoricalAwards=false
• 关键词: Scalar; curvature; geometric; variational; problems

Curvature describes local bending of a space, and is used to distinguish how two shapes are different. This project concerns a particular notion of curvature, called the scalar curvature. Scalar curvature determines 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. Curvature arises throughout the natural sciences, and particularly in general relativity, where scalar curvature is the Lagrangian density of the Einstein-Hilbert action. A natural yet deep question is to understand the effects of scalar curvature on the global properties of a manifold. In particular, the project will focus on geometric variational problems, including minimal surfaces and soap bubbles. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. Some key questions include the existence, regularity and topology of minimal surfaces. These two aspects of the research are deeply related, and will advance our understanding of the shape of nature. The PI will integrate this research with a variety of knowledge-disseminating activities that include organizing seminars, conferences, mini-courses, and reading groups, and giving public lectures. The proposed research concerns a range of topics in differential geometry, geometric measure theory and partial differential equations. Particularly, the PI would like to focus on the following four main topics. The first topic is the investigation of the obstruction problem for manifolds with positive scalar curvature, including the well-known `K(pi, 1) conjecture’. The second topic is to further understand the geometric comparison theorem for scalar curvature lower bound using Riemannian polyhedra. Minimal surface and soap bubbles are key technical tools for such problems. The third topic is to further investigate the stable Bernstein problem for minimal surfaces in R^n. The fourth topic is to further study the moduli space of positive scalar curvature metrics on 3-manifolds (with or without boundary). Understanding such questions has potential applications in 4-dimensional general relativity and in topology.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.

曲率描述了空间的局部弯曲，用于区分两个形状的不同。这个项目涉及一个特殊的曲率概念，称为标量曲率。标量曲率决定了黎曼流形中一个小测地线球的体积与欧氏空间中标准球的体积的偏差。曲率出现在自然科学中，特别是在广义相对论中，标量曲率是爱因斯坦-希尔伯特作用量的拉格朗日密度。一个自然而深刻的问题是理解标量曲率对流形的全局性质的影响。 特别是，该项目将侧重于几何变分问题，包括最小曲面和肥皂泡。极小曲面是自然界中许多界面的数学模型。在广义相对论中，最小表面出现在黑洞的“视视界”;肥皂膜和毛细界面也提供了最小表面的例子。一些关键问题包括极小曲面的存在性、正则性和拓扑性。这两个方面的研究是密切相关的，并将促进我们对自然形状的理解。 PI将把这项研究与各种知识传播活动结合起来，包括组织研讨会、会议、迷你课程、阅读小组和公开讲座。拟议的研究涉及微分几何，几何测度理论和偏微分方程的一系列主题。特别是，PI希望重点关注以下四个主要议题。第一个主题是研究具有正数量曲率的流形的阻塞问题，包括著名的“K（pi，1）曲率”。第二个主题是进一步理解利用黎曼多面体的数量曲率下界的几何比较定理。最小的表面和肥皂泡是解决这些问题的关键技术工具。第三个主题是进一步研究R^n中极小曲面的稳定伯恩斯坦问题。第四个主题是进一步研究三维流形（有边界和无边界）上正数量曲率度量的模空间。理解这些问题在四维广义相对论和拓扑学中有潜在的应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。