Geometric Analysis and Mathematical Relativity Seminar——Some aspects of the minimal surface theory
报告人：Dr. Xiaoxiang Chai (Chinese University of Hong Kong)
时间：2018-07-25 14:30-16:30
地点：Room 1556S, Science Building No. 1
Abstract: A manifold $(M^n, g)$ is called asymptotically flat if outside a compact set it is diffeomorphic to the Euclidean half space minus a ball and its metric $g$ satisfies some decay conditions. We can define a geometric invariant, the ADM mass. The ADM mass is nonnegative provided that the scalar curvature of $M$ and mean curvature of $\partial M$ are both nonnegative; the ADM mass is nonnegative if and only if $M$ is isometric to the standard Euclidean half space. This result is called the positive mass theorem with a noncompact boundary.
Almaraz, Barbosa and de Lima in a recent article use minimal surfaces to prove this theorem. We use minimal surface with free boundary. Formally, this is similar to Schoen and Yau's original proof. We prove that the manifold $T^{n - 1}×[0, 1])#M_0$ does not admit a positive scalar curvature metric with minimal boundary. We use Lohkamp's idea to study the relationship of the geometry of $T^{n - 1}×[0, 1])#M_0$ and the positive mass theorem.
Based on the definition of ADM mass, we derive definitions of Hawking mass and isoperimetric mass with boundary and prove their convergence to the ADM mass.
