A generalization of Liu-Yau's quasi-local mass

Abstract

In [11,12], Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam [18]. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…In addition, in the calculation of the large sphere limit of in asymptotically anti-de Sitter spacetimes it seems natural to choose the reference configuration by embedding into a hyperbolic rather than Euclidean three-space. These issues motivate the following generalization [542] of the Kijowski-Liu-Yau expression.…”

Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds

“…It is important that the estimates in this section do not depend on the area of our initial condition and this requires a careful bookkeeping. Finally, in Section 5 we adapt the work of Shi-Tam [11] and Mu-Tao-Yau [12] to prove long time existence for a flow inspired in [11].…”

On Weyl’s Embedding Problem in Riemannian Manifolds

“…However, for general Riemannian manifold, in lack of conformal Killing vector field, the Minkowski formula no longer exists. In order to bound the total mean curvature, we use the positivity of hyperbolic version of quasi-local mass, which is proved by Wang and Yau [33] and Shi and Tam [32]. The total mean curvature is thus bounded due to the estimates of isometric embedding in hyperbolic space.…”