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The lock principle for scalar curvature

Georg Frenck, Bernhard Hanke, Sven Hirsch

Abstract

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components.

The lock principle for scalar curvature

Abstract

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components.
Paper Structure (3 sections, 2 theorems, 28 equations, 1 figure)

This paper contains 3 sections, 2 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof Theorem \ref{['thm main']}
  3. Concluding remarks

Key Result

Theorem 1

Assume that for $1 \leq i \leq N$, the mean curvatures $H_{i, \mp}$ are strictly positive, and that there exists some $1 \leq \Lambda < N$ such that Furthermore, assume that Then the ADM mass of $( M,g)$ is nonnegative.

Figures (1)

  • Figure 1: The lock principle for $N=4$ and $\Lambda=2$. Two mean-concave curvature jumps at $\Sigma_1$ and $\Sigma_2$ are offset by two mean-convex curvature jumps at $\Sigma_3$ and $\Sigma_4$. The constants $c_i$ determining the symmetric tensor $k$ are defined in \ref{['eq:def-of-c-and-d']}.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : Proof of \ref{['thm main']}