Mass-capacity inequality modeled on conformally flat manifolds

TL;DR

This paper establishes a mass-capacity inequality for certain asymptotically flat manifolds with nonnegative scalar curvature, showing equality characterizes harmonic conformality to Euclidean space minus a set.

Contribution

It introduces a new mass-capacity inequality in the spin case for generalized asymptotically flat manifolds with nonnegative scalar curvature.

Findings

Equality case implies harmonic conformality to Euclidean space minus a set.

The set removed has Hausdorff dimension at most (n-2)/2.

Provides a geometric characterization related to mass and capacity.

Abstract

In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.