On geometric problems related to Brown-York and Liu-Yau quasilocal mass.(English)Zbl 1200.58018

Summary: We discuss some geometric problems related to the definitions of quasilocal mass proposed by J. D. Brown and J. W. York [Contemp. Math. 132, 129–142 (1992; Zbl 0784.53039); Phys. Rev. D (3) 47(4), 1407–1419 (1993)] and C.-C. M. Liu and S.-T. Yau [Phys. Rev. Lett. 90(23), 231102 (2003); J. Am. Math. Soc. 19, No. 1, 181–204 (2006; Zbl 1081.83008)].
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 the 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 two dimensional surfaces evolving in an ambient three dimensional manifold. As a by-product, we are able to write the ADM mass [R. Arnowitt et al. in Phys. Rev., II. Ser. 122, 997–1006 (1961; Zbl 0094.23003)] 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 $${\mathbb {R}^{3,1}}$$ for which the Liu-Yau mass is defined, if $$\Sigma$$ bounds a compact spacelike hypersurface in $${\mathbb {R}^{3,1}}$$, then the Liu-Yau mass of $$\Sigma$$ is strictly positive unless $$\Sigma$$ lies on a hyperplane. We also show that the examples given by N. Ó Murchadha et al. [Phys. Rev. Lett. 92, 259001 (2004)] are special cases of this result.

MSC:

 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 83C99 General relativity 53Z05 Applications of differential geometry to physics
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