On second variation of Wang-Yau quasi-local energy

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.

36 pages

References in corpus (3)

• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Critical points of Wang-Yau quasi-local energy

Cited by in corpus (7)

• Conserved quantities in general relativity: from the quasi-local level to spatial infinity
• Chen-Nester-Tung quasi-local energy and Wang-Yau quasi-local mass
• Quasilocal energy and surface geometry of Kerr spacetime
• Evaluating small sphere limit of the Wang-Yau quasi-local energy
• Energy, momentum, and center of mass in general relativity
• Small sphere limit of the quasi-local energy with anti de-Sitter space reference
• Boundary Effect of Ricci Curvature