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Rigidity of spin fill-ins with non-negative scalar curvature

Simone Cecchini, Sven Hirsch, Rudolf Zeidler

Abstract

We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative.

Rigidity of spin fill-ins with non-negative scalar curvature

Abstract

We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative.
Paper Structure (11 sections, 20 theorems, 68 equations)

This paper contains 11 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. The boundary Dirac operator and Weitzenböck formula
  4. Generalized APS boundary conditions
  5. Spin maps and local boundary conditions
  6. NNSC fill-ins with non-negative mean curvature
  7. The main integral formula
  8. Almost rigidity for maps to Euclidean domains
  9. Applications to asymptotically flat manifolds
  10. The index theorem
  11. A quantitative matrix Hölder inequality

Key Result

Theorem 1.2

Let $\Sigma$ be a closed spin manifold of dimension $n \geq 3$ with $n \equiv 0,1,3$ or $7 \mod 8$. Then there exists a metric $g_\Sigma$ on $\Sigma$ such that every spin NNSC fill-in of $(\Sigma,g_\Sigma)$ with non-negative mean curvature is Ricci-flat with minimal boundary. Certain Berger spheres

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Example 2.1
  • Example 2.3
  • Lemma 3.1
  • ...and 38 more