The Dirac operator on untrapped surfaces

TL;DR

This paper derives a precise lower bound for the first eigenvalue of the Dirac operator on untrapped surfaces in initial data sets, linking it to mean curvature, and explores implications for rigidity and uniqueness in geometric contexts.

Contribution

It introduces a sharp extrinsic lower bound for the Dirac operator's eigenvalue on untrapped surfaces, with new rigidity and uniqueness results in geometric analysis.

Findings

Established a sharp lower bound for the Dirac operator eigenvalue

Derived rigidity results for constraint equations with spherical boundary

Proved uniqueness of constant mean curvature surfaces in Minkowski space

Abstract

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.