Abstract
In this paper, we prove the following theorem regarding the Wang–Yau quasi-local energy of a spacelike two-surface in a spacetime: Let Σ be a boundary component of some compact, time-symmetric, spacelike hypersurface Ω in a time-oriented spacetime N satisfying the dominant energy condition. Suppose the induced metric on Σ has positive Gaussian curvature and all boundary components of Ω have positive mean curvature. Suppose H ≤ H 0 where H is the mean curvature of Σ in Ω and H 0 is the mean curvature of Σ when isometrically embedded in \({\mathbb R^3}\) . If Ω is not isometric to a domain in \({\mathbb R^3}\), then
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1.
the Brown–York mass of Σ in Ω is a strict local minimum of the Wang–Yau quasi-local energy of Σ.
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2.
on a small perturbation \({\tilde{\Sigma}}\) of Σ in N, there exists a critical point of the Wang–Yau quasi-local energy of \({\tilde{\Sigma}}\)
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Bray H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001)
Brown, J.D., York, J.W. Jr.: Quasilocal energy in general relativity. In: Mathematical aspects of classical field theory (Seattle, WA, 1991), vol. 132 Contemp. Math., pp. 129–142. American Mathematical Society, Providence (1992)
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)
Chen, P.-N., Wang, M.-T., Yau, S.-T.: Evaluating quasilocal energy and solving optimal embedding equation at null infinity. arXiv:1002.0927v2
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Huang L.-H.: On the center of mass of isolated systems with general asymptotics. Class. Quantum. Grav. 26, 015012 (2009)
Huisken G., Ilmanen T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)
Lam, M.-K.: The graphs cases of the Riemannian positive mass and Penrose inequalities in all dimensions. arXiv:1010.4256
Li, P.: Lecture notes on geometric analysis. Lecture Notes Series, 6. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul. http://math.uci.edu/~pli/ (1993)
Liu C.-C.M., Yau S.-T.: Positivity of quasilocal mass. Phys. Rev. Lett. 90(23), 231102 (2003)
Liu C.-C.M., Yau S.-T.: Positivity of quasilocal mass II. J. Am. Math. Soc. 19(1), 181–204 (2006)
Miao P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2003)
Miao P.: On a localized Riemannian Penrose inequality. Commun. Math. Phys. 292(1), 271–284 (2009)
Miao P., Shi Y.G., Tam L.-F.: On geometric problems related to Brown–York and Liu–Yau quasilocal mass. Commun. Math. Phys. 298, 437–459 (2010)
Nirenberg L.: The Weyl and Minkowski problems in differential geoemtry in the large. Commun. Pure Appl. Math. 6, 337–394 (1953)
Reilly R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26(3), 459–472 (1977)
Shi Y.-G., Tam L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002)
Wang M.-T., Yau S.-T.: Quasilocal mass in general relativity. Phys. Rev. Lett. 102, 021101 (2009)
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)
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Communicated by Piotr T. Chrusciel.
P. Miao’s research was partially supported by Australian Research Council Discovery Grant #DP0987650.
L.-F. Tam’s research was partially supported by Hong Kong RGC General Research Fund #CUHK 403108.
N. Xie’s research was partially supported by the National Science Foundation of China #10801036, #11011140233 and the Innovation Program of Shanghai Municipal Education Commission #11ZZ01.
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Miao, P., Tam, LF. & Xie, N. Critical Points of Wang–Yau Quasi-Local Energy. Ann. Henri Poincaré 12, 987–1017 (2011). https://doi.org/10.1007/s00023-011-0097-0
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DOI: https://doi.org/10.1007/s00023-011-0097-0