Limit of quasilocal mass integrals in asymptotically hyperbolic manifolds

Kwong, Kwok-Kun; Tam, Luen-Fai

doi:10.1090/S0002-9939-2012-11294-8

Limit of quasilocal mass integrals in asymptotically hyperbolic manifolds
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by Kwok-Kun Kwong and Luen-Fai Tam

Proc. Amer. Math. Soc. 141 (2013), 313-324

DOI: https://doi.org/10.1090/S0002-9939-2012-11294-8

Published electronically: May 3, 2012

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Abstract:

In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well-known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an asymptotically flat manifold.

References

Lars Andersson and Mattias Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998), no. 1, 1–27. MR 1616570, DOI 10.1023/A:1006547905892
Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. MR 2038048, DOI 10.2140/pjm.2003.212.231
Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR 0394451
Xu-Qian Fan and Kwok-Kun Kwong, The Brown-York mass of revolution surfaces in asymptotically Schwarzschild manifolds, J. Geom. Anal. 21 (2011), no. 3, 527–542. MR 2810842, DOI 10.1007/s12220-010-9157-9
Xu-Qian Fan, Yuguang Shi, and Luen-Fai Tam, Large-sphere and small-sphere limits of the Brown-York mass, Comm. Anal. Geom. 17 (2009), no. 1, 37–72. MR 2495833, DOI 10.4310/CAG.2009.v17.n1.a3
Ralph Howard, Blaschke’s rolling theorem for manifolds with boundary, Manuscripta Math. 99 (1999), no. 4, 471–483. MR 1713810, DOI 10.1007/s002290050186
Yanyan Li and Gilbert Weinstein, A priori bounds for co-dimension one isometric embeddings, Amer. J. Math. 121 (1999), no. 5, 945–965. MR 1713298
Pengzi Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182 (2003). MR 1982695, DOI 10.4310/ATMP.2002.v6.n6.a4
A. V. Pogorelov, Some results on surface theory in the large, Advances in Math. 1 (1964), no. fasc. 2, 191–264. MR 178438, DOI 10.1016/0001-8708(65)90039-3
Yuguang Shi and Luen-Fai Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. MR 1987378
Yuguang Shi and Luen-Fai Tam, Rigidity of compact manifolds and positivity of quasi-local mass, Classical Quantum Gravity 24 (2007), no. 9, 2357–2366. MR 2320089, DOI 10.1088/0264-9381/24/9/013
Yuguang Shi, Guofang Wang, and Jie Wu, On the behavior of quasi-local mass at the infinity along nearly round surfaces, Ann. Global Anal. Geom. 36 (2009), no. 4, 419–441. MR 2562923, DOI 10.1007/s10455-009-9169-5
A. Treibergs, Private communications.
Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. MR 1879228
Mu-Tao Wang and Shing-Tung Yau, A generalization of Liu-Yau’s quasi-local mass, Comm. Anal. Geom. 15 (2007), no. 2, 249–282. MR 2344323
Xiao Zhang, A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds. I, Comm. Math. Phys. 249 (2004), no. 3, 529–548. MR 2084006, DOI 10.1007/s00220-004-1056-0