Scalar curvature and the relative capacity of geodesic balls

TL;DR

This paper demonstrates that scalar curvature in a Riemannian manifold can be determined by the relative capacities of small geodesic balls, linking geometric analysis with concepts from general relativity.

Contribution

It introduces the use of relative capacities of geodesic balls to recover scalar curvature, extending previous volume-based methods and proposing a new characterization of flatness.

Findings

Scalar curvature is determined by relative capacities of small geodesic balls.

The result connects geometric analysis with general relativity.

Proposes a conjecture relating capacity behavior to flatness.

Abstract

In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), regarding whether Euclidean-like behavior of the relative capacity on the small scale is sufficient to characterize a space as flat.