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The equivalence of isocapacitary notions of mass

Luca Benatti

TL;DR

The paper proves that a family of isocapacitary masses $ ext{ma}_{ ext{iso}}^{p}$, defined via $p$-capacity, coincides with Huisken's isoperimetric mass $ ext{ma}_{ ext{iso}}$ for all $p eq 1$ under nonnegative scalar curvature and a Euclidean isoperimetric inequality. It shows $ ext{ma}_{ ext{iso}}$ is the largest among the iso-$p$ masses (via a weak $p$-IMCF and asymptotic isoperimetric control) and also the smallest (via Hawking mass monotonicity and capacity bounds, including Xiao–Bray-type results). The $p=1$ case is handled separately with a direct comparison between capacity and perimeter. Together, these results establish the full equivalence $ ext{ma}_{ ext{iso}}^{p}= ext{ma}_{ ext{iso}}$ for all $p\uparrow 3$, broadening the invariant interpretation of geometric mass in nonnegative scalar curvature settings.

Abstract

In this short note, we will prove the equivalence of the isocapacitary notions of mass. This family also includes G. Huisken's isoperimetric mass and J. L. Jauregui's isocapacitary mass.

The equivalence of isocapacitary notions of mass

TL;DR

The paper proves that a family of isocapacitary masses , defined via -capacity, coincides with Huisken's isoperimetric mass for all under nonnegative scalar curvature and a Euclidean isoperimetric inequality. It shows is the largest among the iso- masses (via a weak -IMCF and asymptotic isoperimetric control) and also the smallest (via Hawking mass monotonicity and capacity bounds, including Xiao–Bray-type results). The case is handled separately with a direct comparison between capacity and perimeter. Together, these results establish the full equivalence for all , broadening the invariant interpretation of geometric mass in nonnegative scalar curvature settings.

Abstract

In this short note, we will prove the equivalence of the isocapacitary notions of mass. This family also includes G. Huisken's isoperimetric mass and J. L. Jauregui's isocapacitary mass.

Paper Structure

This paper contains 5 sections, 9 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The case p=1
  3. The isoperimetric mass is the biggest
  4. The isoperimetric mass is the smallest
  5. Final remarks

Key Result

Theorem 2

Let $(M,g)$ be a Riemannian $3$-manifold with nonnegative scalar curvature. Suppose that $M$ possibly has a smooth compact minimal boundary and no other compact minimal surface is contained in $M$. Assume that $M$ satisfies an Euclidean isoperimetric inequality, namely Then, $\ma^{ p}_{\iso} = \ma_{\iso}$ for all $p\in [1,3)$.

Theorems & Definitions (11)

  • Definition 1: Isocapacitary masses
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Definition 5
  • Lemma 6
  • Proposition 7
  • Theorem 8: jauregui_lower_2019
  • Lemma 9
  • Lemma 10
  • ...and 1 more