Geometric Problems Involving Scalar Curvature

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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.