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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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References:

[1] Arnowitt R., Deser S., Misner C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. (2) 122, 997–1006 (1961) · Zbl 0094.23003
[2] Bartnik, R.: Private communications
[3] Bartnik R.: Regularity of variational maximal surfaces. Acta Math. 161(3–4), 145–181 (1988) · Zbl 0667.53049
[4] Bartnik R., Simon L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152 (1982) · Zbl 0512.53055
[5] Bray, H., Khuri, M.: P.D.E.’s which imply the Penrose Conjecture. http://arxiv.org/abs/0905.2622v1[math.DG] , 2009 · Zbl 1244.83016
[6] Brown, J. D., York, J.W. Jr.: Quasilocal energy in general relativity. In: Mathematical aspects of classical field theory (Seattle, WA, 1991), Volume 132 of Contemp. Math., Providence, RI: American Mathematical Society, 1992, pp. 129–142 · Zbl 0784.53039
[7] Brown J.D., York J.W. Jr: Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47(4), 1407–1419 (1993)
[8] Corvino J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000) · Zbl 1031.53064
[9] Fan X.-Q., Shi Y.-G., Tam L.-F.: Large-sphere and small-sphere limits of the Brown-York mass. Comm. Anal. Geom. 17, 37–72 (2009) · Zbl 1175.53083
[10] Flatherty F.J.: The boundary value problem for maximal hypersurfaces. Proc. Natl. Acad. Sci. USA. 76(10), 4765–4767 (1979) · Zbl 0428.49031
[11] Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Second edition. Springer-Verlag, Berlin-Heidelberg-NewYork (1983) · Zbl 0562.35001
[12] Huisken G., Ilmanen T.: The inverse mean curvature flow and the Riemannian Penrose Inequality. J. Diff. Geom. 59, 353–437 (2001) · Zbl 1055.53052
[13] Kobayashi O.: A differential equation arising from scalar curvature function. J. Math. Soc. Japan 34(4), 665–675 (1982) · Zbl 0495.53038
[14] Liu C.-C.M., Yau S.-T.: Positivity of quasilocal mass. Phys. Rev. Lett. 90(23), 231102 (2003) · Zbl 1267.83028
[15] Liu C.-C.M., Yau S.-T.: Positivity of quasilocal mass II. J. Am. Math. Soc. 19(1), 181–204 (2006) · Zbl 1081.83008
[16] Miao P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2002)
[17] Miao P., Tam L.-F.: On the volume functional of compact manifolds with boundary with constant scalar curvature. Calc. Var. Part. Diff. Eq. 36(2), 141 (2009) · Zbl 1175.49043
[18] Nirenberg L.: The Weyl and Minkowski problems in differential geoemtry in the large. Comm. Pure Appl. Math. 6, 337–394 (1953) · Zbl 0051.12402
[19] N. Ó. Murchadha: The Liu-Yau mass as a quasi-local energy in general relativity. http://arxiv.org/abs/0706.1166v1[gr-qc] , 2007
[20] Murchadha N.Ó., Szabados L.B., Tod K.P.: Comment on ”Positivity of quasilocal mass”. Phys. Rev. Lett. 92, 259001 (2004) · Zbl 1267.83029
[21] Shi Y.-G., Tam L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62, 79–125 (2002) · Zbl 1071.53018
[22] Shi Y.-G., Tam L.-F.: Rigidity of compact manifolds and positivity of quasi-local mass. Class. Quant. Grav. 24(9), 2357–2366 (2007) · Zbl 1115.83006
[23] Christodoulou D., Yau S.-T. (1986) Some remarks on the quasi-local mass. In: Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, Providence, RI: Amer. Math. Soc., 1986, pp. 9–14
[24] Wang M.-T., Yau S.-T.: A generalization of Liu-Yau’s quasi-local mass. Comm. Anal. Geom. 15(2), 249–282 (2007) · Zbl 1171.53336
[25] Wang M.-T., Yau S.-T.: Isometric embeddings into the Minkowski space and new quasi-local mass. Commun. Math. Phys. 288(3), 919–942 (2009) · Zbl 1195.53039
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