General Relativity Seminar
Wednesday, March 24, 2021, 4:00pm, Zoom link
Yiyue Zhang (Duke University, Mathematics)
The hyperbolic positive mass theorem and volume comparison involving scalar curvature

Abstract:

In this talk, I will introduce the harmonic level set method developed by Stern in 2019. This technique has been used to prove the positive mass theorems in various settings, for example, the Riemannian case, the spacetime case, the hyperbolic case, and the positive mass theorem with charge. I will focus on the positive mass theorem for asymptotically hyperbolic manifolds. We give a lower bound for the mass in the asymptotically hyperbolic setting. In this setting, we solve the spacetime harmonic equation and give an explicit expansion for the solution. We also prove some rigidity results as corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri. In addition, I will introduce some volume comparison theorems involving scalar curvature. In dimension 3, we have Bray's football theorem which is a sharp volume comparison theorem involving scalar curvature. In higher dimensions, some special cases can be proven.

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