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Renormalized volume and the evolution of APEs

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EN
Abstrakty
EN
We study the evolution of the renormalized volume functional for even-dimensional asymptotically Poincaré-Einstein metrics (M, g) under normalized Ricci flow. In particular, we prove that [...] where S(g(t)) is the scalar curvature for the evolving metric g(t). This implies that if S +n(n − 1) ≥ 0 at t = 0, then RenV(Mn , g(t)) decreases monotonically. For odd-dimensional asymptotically Poincaré-Einstein metrics with vanishing obstruction tensor,we find that the conformal anomaly for these metrics is constant along the flow. We apply our results to the Hawking-Page phase transition in black hole thermodynamics.We compute renormalized volumes for the Einstein 4-metrics sharing the conformal infinity of an AdS-Schwarzschild black hole. We compare these to the free energies relative to thermal hyperbolic space, as originally computed by Hawking and Page using a different regularization technique, and find that they are equal.
Wydawca
Czasopismo
Rocznik
Tom
1
Numer
1
Opis fizyczny
Daty
otrzymano
2015-09-22
zaakceptowano
2015-10-15
online
2015-12-07
Twórcy
autor
  • Department of Mathematics, Seattle University, 901 12th Ave, Seattle, WA 98122
autor
  • Department of Mathematics, Stanford University, Stanford, CA 94305
autor
  • Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton,
    Alberta, T6G 2G1, Canada
Bibliografia
  • [1] P. Albin, Renormalizing curvature integrals on Poincaré-Einstein manifolds, Adv. Math. 221 (2009) 140–169.[WoS]
  • [2] P. Albin, C. Aldana and F. Rochon, Ricci flow and the determinant of the Laplacian on non-compact surfaces, Comm. Par. Diff.Eq. 38 (2013) 711–749.
  • [3] S. Alexakis and R. Mazzeo, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds Comm.Math. Phys. 297 (2010) No. 3, pp. 621–651.[WoS]
  • [4] M.T. Anderson, L2 curvature and volume renormalization of AHE metrics on 4-manifolds, Math. Res. Lett. 8 (2001) 171–188.[Crossref]
  • [5] E. Bahuaud, Ricci flow of conformally compact metrics, Ann. Inst. H. Poincaré: Anal. Non-Lin. 28 (2011) 813–835.[Crossref]
  • [6] T. Balehowsky and E. Woolgar, The Ricci flow of asymptotically hyperbolic mass and applications, J. Math. Phys. 53 (2012)072501.[WoS]
  • [7] S. Brendle and O. Chodosh, A volume comparison theorem for asymptotically hyperbolic manifolds, Comm. Math. Phys., 332(2014) 839–846.[WoS]
  • [8] M. Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. ÉNS 4e série 3 (1970) 285–294.
  • [9] S.-Y.A. Chang, H. Fang and C.R. Graham A note on renormalized volume functionals, preprint [arXiv:1211.6422].[WoS]
  • [10] S.-Y.A. Chang, J. Qing and P. Yang, On the renormalized volumes for conformally compact Einstein manifolds, J. Math. Sci.149 (2008) 1755–1769.
  • [11] S. de Haro, K. Skenderis, and S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFTcorrespondence, Comm. Math. Phys. 217 (2001) 595–622.
  • [12] K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991) 547–569.
  • [13] C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les Mathématiques d’Aujourd’hui, Astérisque (numérohors série, 1985) 95–116.
  • [14] C. Fefferman and C.R. Graham, The ambient metric Princeton Annals of Mathematical Studies 178, Princeton, NJ (2012)
  • [15] D.H. Friedan, Nonlinear Models in 2 + " Dimensions, PhD thesis, University of California, Berkeley, 1980 (unpublished);Phys. Rev. Lett. 45 (1980) 1057–1060; Ann. Phys. (NY) 163 (1985) 318–419.[Crossref]
  • [16] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics in The Proceedings of the 19thWinter School "Geometry and Physics” (Srni, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63 (2000), 31–42.
  • [17] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Q-curvature, in AdS/CFT Correspondence: Einstein Metricsand Their Conformal Boundaries, IRMA Lect Math Theor Phys 8 (European Mathematical Society Zürich, 2005) pp 59–71.
  • [18] C.R. Graham and J. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991) 186–225.
  • [19] C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B546(1999) 52–64.
  • [20] S.W. Hawking and D.N. Page, Thermodynamics of black holes in anti-de Sitter space, Comm. Math. Phys. 87 (1983) 577–588.
  • [21] M. Headrick and T. Wiseman, Ricci flow and black holes, Class. Quantum Grav. 23 (2006) 6683–6708.
  • [22] M. Henningson and K. Skenderis, The holographic Weyl anomaly, J.H.E.P. 9807 (1998) 023.
  • [23] X. Hu, D. Ji, and Y. Shi, Volume comparison of conformally compact manifolds with scalar curvature R ≥ −n(n − 1).http://arxiv.org/abs/1309.5430.
  • [24] J. Isenberg, R. Mazzeo, and N. Sesum, Ricci flow on asymptotically conical surfaces with nontrivial topology, J. Reine Angew.Math. (Crelle), 676 (2013) 227–248.[WoS]
  • [25] K Krasnov and J-M Schlenker, On the renormalized volume of hyperbolic 3-manifolds, Commun Math Phys 279 (2008) 637–668.[WoS]
  • [26] J.M.Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.Math. Phys. 2 (1998) 231–252.
  • [27] T.A. Oliynyk and E. Woolgar, Asymptotically flat Ricci flows, Commun. Anal. Geom. 15 (2007) 535–568.[Crossref]
  • [28] I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, in AdS/CFT correspondence: Einstein metrics andtheir conformal boundaries, IRMA Lectures in Mathematics and Theoretical Physics 8, pp 73–101, European MathematicalSociety, Zürich (2005).
  • [29] T. Prestidge, Dynamic and thermodynamic stability and negative modes in Schwarzschild-anti-de Sitter, Phys. Rev. D61 (2000)084002.
  • [30] J. Qing, On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 21 (2003) 1141–1153.[Crossref]
  • [31] J. Qing, Y. Shi, and J. Wu, Normalized Ricci flows and conformally compact Einstein metrics, Calc. Var. Partial DifferentialEquations 46 (2013), no. 1-2, 183–211.[WoS]
  • [32] K. Schleich, S. Surya, and D.M. Witt, Phase transitions for flat adS black holes, Phys Rev Lett 86 (2001) 5231–5234.
  • [33] W.X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff. Geom. 30 (1989) 223–301.
  • [34] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998)505–532.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_geofl-2015-0007
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