On Geometric Problems Related to Brown-York and Liu-Yau Quasilocal Mass

Abstract

Abstract. We discuss some geometric problems related to the definitions of quasilocal mass proposed by [6] and Liu-Yau [13] [14]. 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 [1] 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 Φ(x) in the asymptotic region outside S r . In the third part, we prove that for any closed, spacelike, 2-surface Σ in the Minkowski space R 3,1 for which the Liu-Yau mass is defined, if Σ bounds a compact spacelike hypersurface in R 3,1 , then the Liu-Yau mass of Σ is strictly positive unless Σ lies on a hyperplane. We also show that the examples given byÓ Murchadha, Szabados and Tod [18] are special cases of this result.

The Dirac Operator on Untrapped Surfaces

“…Proof of Theorem 8: From Lemma 4.1 in [15], Σ spans a compact, smoothly immersed, maximal hypersurface Ω in R 3,1 . Denote by K the second fundamental form of Ω in R 3,1 (which satisfies Tr(K) = 0 since Ω is maximal), the Gauss formula gives R = |K| 2 ≥ 0.…”

“…Moreover, since this estimate only involves geometrical data of the surface Σ and does not depend on the bounding domain Ω, we focus especially on the geometrical properties of the surfaces for which (2) is an equality. More precisely, following[15], we prove…”

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

“…Indeed, specific two-surfaces in the Minkowski spacetime are given in [401], for which . Moreover, it is shown in [361] that the Kijowski-Liu-Yau energy for a closed two-surface in Minkowski spacetime strictly positive unless lies in a spacelike hyperplane. On the applicability of in the formulation and potential proof of Thorne’s hoop conjecture see Section 13.2.2.…”

Some functionals on compact manifolds with boundary

“…where H(g) is the usual mean curvature of Σ in (Ω, g), it was proved in [10] that g 0 is a critical point of E…”