Abstract
The aim of this project is to apply the power and versatility of optimal transport to the study of the almost rigidity of the positive mass theorem. The positive mass theorem is motivated by general relativity, and it states that asymptotically flat Riemannian manifolds with non-negative scalar curvature must have non-negative mass. Furthermore, the only asymptotically flat Riemannian manifold with non-negative scalar curvature and zero mass is Euclidean space. It is natural to ask if the positive mass theorem satisfies almost rigidity: if an asymptotically flat Riemannian manifold has small mass, is it true that it must be close to Euclidean space? The appropriate setting for this question is non-smooth Riemannian geometry, and metrics on the space of Riemannian manifolds. Optimal transport has been extremely successful in these areas, and so it is natural to attempt to apply its techniques to the almost rigidity of the positive mass theorem. However, this has not been done yet. In this project, we propose to do the following: 1) Study the almost rigidity of the Riemannian positive mass theorem using metrics arising from optimal transport. 2) Relate metrics currently being used to study the almost rigidity of the positive mass theorem with metrics arising from optimal transport.
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