Mass-capacity inequalities for conformally flat manifolds with boundary

TL;DR

This paper establishes new mass-capacity and Penrose inequalities for conformally flat manifolds across dimensions, also deriving classical geometric inequalities as corollaries, with equality cases fully characterized.

Contribution

It introduces novel mass-capacity and volumetric Penrose inequalities for conformally flat manifolds, extending geometric analysis in arbitrary dimensions.

Findings

Proved a mass-capacity inequality for conformally flat manifolds.

Established a volumetric Penrose inequality in arbitrary dimensions.

Derived Pólya-Szegő and Aleksandrov-Fenchel inequalities as corollaries.

Abstract

In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, P\'olya-Szeg\"o and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is characterized.