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

TL;DR

This paper proves new rigidity theorems for spin fill-ins with non-negative scalar curvature using advanced spinorial methods, addressing questions by Miao and Gromov, and introduces a novel mass inequality for asymptotically Schwarzschild manifolds.

Contribution

It introduces two novel spinorial techniques for establishing mean curvature rigidity and derives a new mass inequality for asymptotically Schwarzschild manifolds.

Findings

Established new mean curvature rigidity theorems for spin fill-ins.

Derived a Witten-type integral inequality for the mass of asymptotically Schwarzschild manifolds.

Connected spinorial methods to geometric inequalities in scalar curvature geometry.

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.