Lessons from General Relativity for Riemannian Geometry

17.04.2019, 14:00  –  Haus 9, Raum 2.22
Institutskolloquium

Anna Sakovich (Uppsala University), Michael Eichmair (University of Vienna)

14:00 Anna Sakovich (Uppsala University): Mass in Riemannian Geometry and General Relativity

15:00 Tea and Coffee break

15:30 Michael Eichmair (University of Vienna): The isoperimetric problem for asymptotically Euclidean spaces

Abstracts:

Anna Sakovich (Uppsala University): Mass in Riemannian Geometry and General Relativity
Suppose that we are given two Riemannian manifolds that 'look alike' near infinity. Under which minimal geometric assumptions can we determine if these two manifolds are identical or different? If one of the two manifolds is either Euclidean space or hyperbolic space, the answer to this question can be provided by an invariant called mass, which was first discovered in General Relativity. In this talk we will discuss geometry and physics behind this rather unique invariant and review some geometric inequalities where it appears. We will also address some consequences of having small mass, both for Riemannian manifolds and spacetimes.

Michael Eichmair (University of Vienna): The isoperimetric problem for asymptotically Euclidean spaces
A small geodesic ball at a point of positive scalar curvature has more volume than a Euclidean ball with the same perimeter. In fact, the magnitude of the scalar curvature can be computed as an isoperimetric deficit of the geodesic ball. This classical observation has a global counterpart that has recently been established in joint work with O. Chodosh, Y. Shi, and H. Yu: Let (M, g) be a Riemannian 3-manifold whose geometry is asymptotic to that of Euclidean space. We also assume that the scalar curvature of (M, g) is non-negative and that (M, g) is not flat. For every sufficiently large amount of area, there is a unique region of largest volume whose perimeter has that area. These large solutions of the isoperimetric problem are nested and their isoperimetric deficit from Euclidean space encodes the “mass" of (M,g). The goal of my lecture is to explain this effective version of the positive mass theorem and its relation to a conjecture of R. Schoen recently proven in joint work with O. Chodosh.