On the long neck principle and width estimates for initial data sets

TL;DR

This paper establishes the long neck principle and width estimates for initial data sets in general relativity using spinorial methods, providing new geometric inequalities under energy conditions.

Contribution

It introduces a novel spinorial approach to prove the long neck principle and width inequalities for initial data sets in Einstein's equations.

Findings

Proved the long neck principle for initial data sets.

Derived width estimates for geodesic collar neighborhoods.

Established width inequalities under scalar curvature bounds.

Abstract

In this paper, we prove the long neck principle, band width estimates, and width inequalities of the geodesic collar neighborhoods of the boundary in the setting of general initial data sets for the Einstein equations, subject to certain energy conditions corresponding to the lower bounds of scalar curvature on Riemannian manifolds. Our results are established via the spinorial Callias operator approach.