Renormalized volume and the evolution of APEs

Eric Bahuaud; Rafe Mazzeo; Eric Woolgar

Geometric Flows (2015)

  • Volume: 1, Issue: 1
  • ISSN: 2353-3382

Abstract

top
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.

How to cite

top

Eric Bahuaud, Rafe Mazzeo, and Eric Woolgar. "Renormalized volume and the evolution of APEs." Geometric Flows 1.1 (2015): null. <http://eudml.org/doc/276424>.

@article{EricBahuaud2015,
abstract = {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.},
author = {Eric Bahuaud, Rafe Mazzeo, Eric Woolgar},
journal = {Geometric Flows},
keywords = {Ricci flow; conformally compact metrics; asymptotically hyperbolic metrics; renormalized volume; black hole thermodynamics},
language = {eng},
number = {1},
pages = {null},
title = {Renormalized volume and the evolution of APEs},
url = {http://eudml.org/doc/276424},
volume = {1},
year = {2015},
}

TY - JOUR
AU - Eric Bahuaud
AU - Rafe Mazzeo
AU - Eric Woolgar
TI - Renormalized volume and the evolution of APEs
JO - Geometric Flows
PY - 2015
VL - 1
IS - 1
SP - null
AB - 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.
LA - eng
KW - Ricci flow; conformally compact metrics; asymptotically hyperbolic metrics; renormalized volume; black hole thermodynamics
UR - http://eudml.org/doc/276424
ER -

References

top
  1. [1] P. Albin, Renormalizing curvature integrals on Poincaré-Einstein manifolds, Adv. Math. 221 (2009) 140–169. [WoS] Zbl1170.53017
  2. [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. Zbl1284.53060
  3. [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] Zbl1193.53131
  4. [4] M.T. Anderson, L2 curvature and volume renormalization of AHE metrics on 4-manifolds, Math. Res. Lett. 8 (2001) 171–188. [Crossref] Zbl0999.53034
  5. [5] E. Bahuaud, Ricci flow of conformally compact metrics, Ann. Inst. H. Poincaré: Anal. Non-Lin. 28 (2011) 813–835. [Crossref] Zbl1235.53066
  6. [6] T. Balehowsky and E. Woolgar, The Ricci flow of asymptotically hyperbolic mass and applications, J. Math. Phys. 53 (2012) 072501. [WoS] Zbl1279.83034
  7. [7] S. Brendle and O. Chodosh, A volume comparison theorem for asymptotically hyperbolic manifolds, Comm. Math. Phys., 332 (2014) 839–846. [WoS] Zbl1306.53021
  8. [8] M. Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. ÉNS 4e série 3 (1970) 285–294. Zbl0204.54802
  9. [9] S.-Y.A. Chang, H. Fang and C.R. Graham A note on renormalized volume functionals, preprint [arXiv:1211.6422]. [WoS] 
  10. [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. [11] S. de Haro, K. Skenderis, and S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217 (2001) 595–622. Zbl0984.83043
  12. [12] K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991) 547–569. Zbl0707.53008
  13. [13] C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les Mathématiques d’Aujourd’hui, Astérisque (numéro hors série, 1985) 95–116. Zbl0602.53007
  14. [14] C. Fefferman and C.R. Graham, The ambient metric Princeton Annals of Mathematical Studies 178, Princeton, NJ (2012) Zbl1243.53004
  15. [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. [16] C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics in The Proceedings of the 19th Winter School "Geometry and Physics” (Srni, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63 (2000), 31–42. 
  17. [17] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Q-curvature, in AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, IRMA Lect Math Theor Phys 8 (European Mathematical Society Zürich, 2005) pp 59–71. Zbl1074.53027
  18. [18] C.R. Graham and J. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991) 186–225. Zbl0765.53034
  19. [19] C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B546 (1999) 52–64. Zbl0944.81046
  20. [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. [21] M. Headrick and T. Wiseman, Ricci flow and black holes, Class. Quantum Grav. 23 (2006) 6683–6708. Zbl1114.83007
  22. [22] M. Henningson and K. Skenderis, The holographic Weyl anomaly, J.H.E.P. 9807 (1998) 023. Zbl0958.81083
  23. [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. Zbl1338.53071
  24. [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] Zbl1267.53070
  25. [25] K Krasnov and J-M Schlenker, On the renormalized volume of hyperbolic 3-manifolds, Commun Math Phys 279 (2008) 637– 668. [WoS] Zbl1155.53036
  26. [26] J.M.Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor.Math. Phys. 2 (1998) 231–252. Zbl0914.53047
  27. [27] T.A. Oliynyk and E. Woolgar, Asymptotically flat Ricci flows, Commun. Anal. Geom. 15 (2007) 535–568. [Crossref] Zbl1138.53057
  28. [28] I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, in AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lectures in Mathematics and Theoretical Physics 8, pp 73–101, European Mathematical Society, Zürich (2005). Zbl1081.81085
  29. [29] T. Prestidge, Dynamic and thermodynamic stability and negative modes in Schwarzschild-anti-de Sitter, Phys. Rev. D61 (2000) 084002. 
  30. [30] J. Qing, On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 21 (2003) 1141–1153. [Crossref] Zbl1042.53031
  31. [31] J. Qing, Y. Shi, and J. Wu, Normalized Ricci flows and conformally compact Einstein metrics, Calc. Var. Partial Differential Equations 46 (2013), no. 1-2, 183–211. [WoS] Zbl1259.53046
  32. [32] K. Schleich, S. Surya, and D.M. Witt, Phase transitions for flat adS black holes, Phys Rev Lett 86 (2001) 5231–5234. 
  33. [33] W.X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff. Geom. 30 (1989) 223–301. Zbl0676.53044
  34. [34] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505–532. Zbl1057.81550

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.