On geometric problems related to Brown-York and Liu-Yau quasilocal mass

TL;DR

This paper explores geometric aspects of Brown-York and Liu-Yau quasilocal mass, including variational characterizations, derivative formulas, and positivity results, linking these concepts to static metrics and ADM mass in general relativity.

Contribution

It introduces a new variational problem related to Brown-York mass, derives a derivative formula for Brown-York mass evolution, and proves positivity of Liu-Yau mass under certain geometric conditions.

Findings

Critical points of the variational problem are static metrics.

Derived a formula relating ADM mass to Brown-York mass and scalar curvature.

Proved Liu-Yau mass is positive unless the surface is planar.

Abstract

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensional manifold. As an interesting by-product, we are able to write the ADM mass \cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere $S_r$ and an integral of the scalar curvature plus a geometrically constructed function $\Phi(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $\Sigma$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $\Sigma$ bounds a compact spacelike hypersurface in $\R^{3,1}$, then the Liu-Yau mass of $\Sigma$ is strictly positive unless $\Sigma$ lies on a hyperplane. We also show that the examples given by \'{O} Murchadha, Szabados and Tod \cite{MST} are special cases of this result.