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Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli, Marco Pozzetta

Abstract

This paper deals with quasi-local isoperimetric versions of the positive mass theorem on $3$-manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying $C^0$-stable quantitative estimates.

Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

Abstract

This paper deals with quasi-local isoperimetric versions of the positive mass theorem on -manifolds endowed with continuous complete metrics having nonnegative scalar curvature in a suitable weak sense. As a corollary, we derive existence results for isoperimetric sets in such low regularity setting. Our main tool is a new local version of the weak inverse mean curvature flow enjoying -stable quantitative estimates.
Paper Structure (17 sections, 25 theorems, 125 equations, 1 figure)

This paper contains 17 sections, 25 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy
  4. Comments and comparison with related literature
  5. Other notions of weak scalar curvature bounds, and relations with the works PaulaGAFAPaulaADM
  6. The question of rigidity
  7. Preliminaries
  8. Basic definitions and properties of $C^0$-metrics
  9. BV functions and sets of finite perimeter
  10. Technical Lemmas on $C^0$-Riemannian manifolds
  11. Local inverse mean curvature flow
  12. Properties of the weak IMCF
  13. Connectedness of level sets of the weak IMCF
  14. Proof of the main results
  15. Producing a set satisfying the reverse Euclidean isoperimetric inequality
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a complete 3-dimensional $C^0$-Riemannian manifold without boundary. Let $K\subset M$ be a compact set, and let $\mathcal{E}$ be an unbounded connected component $M\setminus K$. Assume that $\mathcal{E}$ is $C^0_{\mathrm{loc}}$-asymptotic to $\mathbb R^3$, see def:C0locasymptotic, and

Figures (1)

  • Figure 1: The picture sketches the assumptions of \ref{['lem:Connectedness']}.

Theorems & Definitions (65)

  • Definition 1: $R_g\geq 0$ in the approximate sense
  • Definition 2: Quasi-local isoperimetric mass
  • Definition 3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 4: $C^0$-Riemannian manifold
  • Lemma 1
  • proof
  • ...and 55 more