On second variation of Wang-Yau quasi-local energy

TL;DR

This paper analyzes the second variation of the Wang-Yau quasi-local energy functional in general relativity, proving its positivity under various geometric conditions on surfaces within 3-manifolds with nonnegative scalar curvature.

Contribution

It establishes the positive definiteness of the second variation functional on large, nearly round, and small geodesic spheres under specific curvature conditions, extending understanding of quasi-local energy stability.

Findings

Functional is positive definite on large coordinate spheres

Functional remains positive on nearly round surfaces including large CMC spheres

Counterexamples show the functional can be negative despite positive Brown-York mass

Abstract

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We also give examples of functions H, which can be made arbitrarily close to the constant 2, on the standard sphere such that the boundary data consisting of the standard spherical metric and H has positive Brown-York mass while the associated functional is negative somewhere.