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Static potentials and area minimizing hypersurfaces

Authors: Lan-Hsuan Huang, Daniel Martin and Pengzi Miao
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 53C21
Published electronically: January 26, 2018
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Abstract: We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must contain a complete non-compact area minimizing hypersurface. Some results related to the Riemannian positive mass theorem, and Bartnik's quasi-local mass are obtained.

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  • [1] Lucas Ambrozio, On static three-manifolds with positive scalar curvature, J. Differential Geom. 107 (2017), no. 1, 1-45. MR 3698233
  • [2] Michael T. Anderson and Jeffrey L. Jauregui, Embeddings, immersions and the Bartnik quasi-local mass conjectures, arXiv:1611.08755 [math.DG].
  • [3] Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661-693. MR 849427,
  • [4] Robert Bartnik, New definition of quasilocal mass, Phys. Rev. Lett. 62 (1989), no. 20, 2346-2348. MR 996396,
  • [5] Robert Beig and Piotr T. Chruściel, Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys. 37 (1996), no. 4, 1939-1961. MR 1380882,
  • [6] Hubert L. Bray, On the positive mass, Penrose, and ZAS inequalities in general dimension, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 1-27. MR 2906919
  • [7] Hubert L. Bray and Piotr T. Chruściel, The Penrose inequality, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 39-70. MR 2098913
  • [8] Simon Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247-269. MR 3090261,
  • [9] Gary L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, Gen. Relativity Gravitation 19 (1987), no. 2, 147-154. MR 876598,
  • [10] Alessandro Carlotto, Otis Chodosh, and Michael Eichmair, Effective versions of the positive mass theorem, Invent. Math. 206 (2016), no. 3, 975-1016. MR 3573977,
  • [11] Piotr T. Chruściel, On analyticity of static vacuum metrics at non-degenerate horizons, Acta Phys. Polon. B 36 (2005), no. 1, 17-26. MR 2125333
  • [12] Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137-189. MR 1794269,
  • [13] Justin Corvino, A note on the Bartnik mass, to appear in Harvard CMSA Lecture Series, International Press, Somerville, MA, 2017.
  • [14] Justin Corvino and Lan-Hsuan Huang, Localized deformation for initial data sets with the dominant energy condition, preprint, arXiv:1606.03078 [math.DG] (2016).
  • [15] Gregory J. Galloway, On the topology of black holes, Comm. Math. Phys. 151 (1993), no. 1, 53-66. MR 1201655
  • [16] Gregory J. Galloway and Pengzi Miao, Variational and rigidity properties of static potentials, Comm. Anal. Geom. 25 (2017), no. 1, 163-183. MR 3663315,
  • [17] Lan-Hsuan Huang, Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics, Comm. Math. Phys. 300 (2010), no. 2, 331-373. MR 2728728,
  • [18] Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353-437. MR 1916951
  • [19] Werner Israel, Event horizons in static electrovac space-times, Comm. Math. Phys. 8 (1968), no. 3, 245-260. MR 1552541,
  • [20] Yu Li, Ricci flow on asymptotically Euclidean manifolds, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-The University of Wisconsin-Madison, 2017. MR 3664958
  • [21] Norman Meyers, An expansion about infinity for solutions of linear elliptic equations, J. Math. Mech. 12 (1963), 247-264. MR 0149072
  • [22] Pengzi Miao, A remark on boundary effects in static vacuum initial data sets, Classical Quantum Gravity 22 (2005), no. 11, L53-L59. MR 2145225,
  • [23] Pengzi Miao and Luen-Fai Tam, Static potentials on asymptotically flat manifolds, Ann. Henri Poincaré 16 (2015), no. 10, 2239-2264. MR 3385979,
  • [24] Pengzi Miao and Luen-Fai Tam, Evaluation of the ADM mass and center of mass via the Ricci tensor, Proc. Amer. Math. Soc. 144 (2016), no. 2, 753-761. MR 3430851,
  • [25] H. Müller zum Hagen, David C. Robinson, and H. J. Seifert, Black holes in static vacuum space-times, General Relativity and Gravitation 4 (1973), 53-78. MR 0398432
  • [26] Jie Qing and Wei Yuan, On scalar curvature rigidity of vacuum static spaces, Math. Ann. 365 (2016), no. 3-4, 1257-1277. MR 3521090,
  • [27] D. C. Robinson, A simple proof of the generalization of Israel's theorem, Gen. Relativity Gravitation 8 (1977), no. 8, 695-698.
  • [28] Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45-76. MR 526976
  • [29] Richard Schoen and Shing Tung Yau, The energy and the linear momentum of space-times in general relativity, Comm. Math. Phys. 79 (1981), no. 1, 47-51. MR 609227
  • [30] Richard Schoen and Shing Tung Yau, Positive scalar curvature and minimal hypersurface singularities, arXiv:1704.05490 (2017).

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Additional Information

Lan-Hsuan Huang
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Daniel Martin
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Pengzi Miao
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146

Received by editor(s): June 21, 2017
Received by editor(s) in revised form: June 24, 2017, August 14, 2017, and August 26, 2017
Published electronically: January 26, 2018
Additional Notes: The first two authors were partially supported by the NSF through grant DMS 1452477.
The third author was partially supported by Simons Foundation Collaboration Grant for Mathematicians #281105.
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society

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