On the local rigidity of Einstein manifolds with convex boundary

TL;DR

This paper proves that under certain convexity and topological conditions, boundary Killing fields extend to the entire Einstein manifold, generalizing classical rigidity results to higher dimensions.

Contribution

It provides a new proof of the infinitesimal rigidity of convex surfaces and extends this to Einstein manifolds of any dimension with convex boundary.

Findings

Killing fields at the boundary extend inward under convexity conditions

The result generalizes classical Euclidean rigidity to Einstein manifolds

A simple fundamental group condition is sufficient for extension

Abstract

Let (M, g) be a compact Einstein manifold with non-empty boundary. We prove that Killing fields at the boundary extend to Killing fields of any (M, g) provided the boundary is weakly convex and a simple condition on the fundamental group holds. This gives a new proof of the classical infinitesimal rigidity of convex surfaces in Euclidean space and generalizes the result to Einstein metrics of any dimension.