Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

TL;DR

This paper introduces nonlinear isocapacitary mass concepts in 3-manifolds with nonnegative scalar curvature, bridging existing mass notions and providing new proofs of fundamental geometric inequalities.

Contribution

It develops a family of nonlinear masses depending on a parameter, unifies them with the ADM mass, and offers a nonlinear potential theoretic proof of the Penrose inequality.

Findings

Masses depend on parameter p, interpolating between known masses.

Positive mass theorems established for these nonlinear masses.

Nonlinear proof of the Penrose inequality provided.

Abstract

We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1<p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.