December 8 (Sun), 2019 11:55--12:55, 15:45--16:45 Kavli Institute for the Physics and Mathematics of the Universe The University of Tokyo, Chiba, Japan |
Abstract
This survey paper centers around the geometry of Riemannian manifolds
of non-negative scalar curvature. This study can be motivated either
from the point of view of general relativity or from a pure Riemannian
geometry point of view. In general relativity such manifolds arise
naturally as spacelike hypersurfaces in physically reasonable spacetimes.
As such there are physical ideas which motivate geometric theorems
on this class of manifolds. In this paper we focus especially on ideas
related to gravitational mass and energy. One focus of the paper is
on the positive mass theorem in general dimensions and its relation
to singularities of volume minimizing hypersurfaces. We also discuss
applications of the positive mass theorem to compactness questions
for metrics of constant scalar curvature, to uniqueness of the Schwarzschild
solution as a static vacuum solution of the Einstein equations, and
to the Penrose inequality relating the area of a horizon to the total mass.
From a purely Riemannian geometry point of view it is natural to study
manifolds of non-negative scalar curvature from either a local or global
viewpoint. We compare local ideas from relativity to polyhedral comparison
theorems from Riemannian geometry.