Geometric Problems Involving Scalar Curvature

广义相对论是爱因斯坦的重力理论，它将重力解释为时空曲率的结果。该理论通过解爱因斯坦方程来预测黑洞的存在。天文学家最近使用事件视界望远镜(EHT)获得的第一张黑洞图像，为这一理论的准确性提供了显著的证据。这一提议的目的是研究广义相对论中空间区域的几何，特别是包括黑洞周围的区域。这个项目的结果将为有限区域内的引力能量以及黑洞对这种准局部能量的贡献提供新的线索。在几何方面，PI旨在研究具有非负标量曲率的有边界的流形。其目的是对标量曲率、边界平均曲率、内部极小曲面、紧致流形的体积和渐近平坦流形的质量之间的相互作用有新的理解。PI还将在具有非负数量曲率的紧致流形的边界上建立一个Poincare类型的不等式。PI将使用变分和偏微分方程中的几何和分析方法来实现这些目标。该奖项反映了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。

General Relativity is Einstein's theory of gravity that interprets gravity as a consequence of the curvature of spacetime. The theory predicts the existence of black holes via solutions to the Einstein equation. The first image of a black hole, recently obtained by astronomers using Event Horizon Telescope (EHT), has given remarkable evidence to the accuracy of this theory. The aim of this proposal is to investigate the geometry of space regions in general relativity, which in particular include regions surrounding a black hole. Results from this project will shed new light on the gravitational energy confined in a finite region, as well as the contribution of black holes to such quasi-local energy.In geometric terms, the PI aims at studying manifolds with non-negative scalar curvature, with boundary. The goal is to obtain new understanding of the interaction among scalar curvature, boundary mean curvature, interior minimal surfaces, volume of compact manifolds, and mass of asymptotically flat manifolds. The PI willl also establish a Poincare-type inequality on the boundary of compact manifolds with non-negative scalar curvature. The PI will employ geometric and analytic methods from calculus of variation and partial differential equations to achieve these goals.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.