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Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds

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Abstract

We study the renormalized area functional \({\mathcal{A}}\) in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3-manifolds (and somewhat more broadly, Poincaré-Einstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formulæ for the first and second variations of \({\mathcal{A}}\) which are given by integrals of global quantities over the asymptotic boundary loop γ of Y. All of these formulæ are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a \({\mathbb{Z}}\) -valued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of \({\mathcal{A}}\) for minimal surfaces in \({\mathbb{H}^3}\) , and finally, discuss the relationship of \({\mathcal{A}}\) to the Willmore functional.

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References

  1. Albin P.: Renormalizing Curvature Integrals on Poincare-Einstein Manifolds. Adv. Math. 221, 140–169 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson M.: Complete Minimal Varieties in hyperbolic space. Invent. Math. 69, 477–494 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Anderson M.: Complete minimal hypersurfaces in hyperbolic n-manifolds. Comment. Math. Helv. 58, 264–290 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anderson M.: Curvature estimates for minimal surfaces in 3-manifolds. Ann. Scient. Éc. Norm. Sup. 4(18), 89–105 (1985)

    Google Scholar 

  5. Anderson M.: L 2 curvature and volume renormalization of AHE metrics on 4-manifolds. Math. Research Lett. 8, 171–188 (2001)

    MATH  Google Scholar 

  6. Anderson, M.: Topics in conformally compact Einstein metrics. In: Perpectives in Riemannian Geometry, Apostolov, V. et al. eds, CRM Proc. Lecture Notes Vol. 40, Providence, RI: Amer. Math. Soc., 2006, pp. 1–26

  7. Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10, 553–576 (2003)

    Article  MathSciNet  Google Scholar 

  8. Biquard, O.: Asymptotically symmetric Einstein metrics. Translated by S Wilson, SMF/AMS Texts and Monographs 13, Providence, RI: Am. Math. Soc., 2006, orig. in French as Asterisque 265, (2000)

  9. Chang, S.Y.-A., Qing, J., Yang, P.: On the renormalized volumes for conformally compact Einstein manifolds. to appear in Proceeding conference in Geometric Analysis, Moscow 2005

  10. Chang, S.Y.-A., Chen, S., Yang, P.: In preparation

  11. Chrusćiel P.T., Delay E., Lee J.M., Skinner D.: Boundary regularity of conformally compact Einstein metrics. J. Diff. Geom. 69, 111–136 (2005)

    MATH  Google Scholar 

  12. Coskunuzer B.: Minimal planes in hyperbolic space. Comm. Anal. Geom. 12(4), 821–836 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Coskunuzer B.: Generic uniqueness of least area planes in hyperbolic space. Geom. and Top. 10, 401–412 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Drukker N., Gross D., Ooguri H.: Wilson Loops and Minimal Surfaces. Phys. Rev. D (3) 60(12), 125006 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  15. Epstein, C.: Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Unpublished manuscript (1983), available at http://www.math.upenn.edu/~cle/papers/index.html

  16. Fefferman, C., Graham, C.R.: Conformal Invariants. Élie Cartan et les mathématiques d’Aujourd’hui, Astérisque, 1985, Numero Hors’serie, 95–116 (1985)

  17. Fefferman, C., Graham, C.R.: The ambient metric. http://arXiv.org/abs/0710.0919v2[math.DG], 2008

  18. Graham, C.R.: Volume and Area Renormalizations for Conformally Compact Einstein Metrics. In: Proc. of 19th Winter School in Geometry and Physics, (Srni, Czech Rep., 1999), Rend. Circ. Mat. Palermo (2) Supp. No. 63, 31–42 (2000)

  19. Graham C.R., Lee J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Graham C.R., Witten E.: Conformal Anomaly Of Submanifold Observables In AdS/CFT Correspondence. Nucl.Phys. B 546, 52–64 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Hardt R., Lin F.: Regularity at infinity for area minimizing hypersurfaces in hyperbolic space. Invent. Math. 88, 217–224 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Henningson M., Skenderis K.: The holographic Weyl anomaly. J. High Energy Phys. 07, 023 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  23. Hirata T., Takayanagi T.: AdS/CFT and Strong Subadditivity of Entanglement Entropy. J. High Energy Phys. 02, 042 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  24. Krasnov K., Schlenker J.M.: On the renormalized volume of hyperbolic 3-manifolds. Commun. Math. Phys 279(3), 637–668 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Lee, J.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Amer. Math. Soc. 183, no. 864, Providence, RI: Amer. Math. Soc., 2006

  26. Lin F.H.: On the Dirichlet problem for the minimal graphs in hyperbolic space. Invent. Math. 96, 593–612 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Maldacena J.: Wilson loops in Large N field theories. Phys. Rev. Lett. 80, 4859 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Mazzeo R.: Elliptic theory of differential edge operators, I. Commun. Par. Diff. Eq. 16(10), 1616–1664 (1991)

    MathSciNet  Google Scholar 

  29. Mazzeo R.: Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds. Amer. J. Math. 113(1), 25–45 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. de Oliveira G., Soret M.: Complete minimal surfaces in hyperbolic space. Math. Annal. 311, 397–419 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Patterson S.J., Perry P.: The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. Duke Math. J. 106(2), 321–390 (2001)

    MATH  MathSciNet  Google Scholar 

  32. Polyakov A.M.: Fine structure of strings. Nucl. Phys. B 268(2), 406–412 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  33. Polyakov A.M., Rychkov V.S.: Loop dynamics and AdS/CFT correspondence. Nucl. Phys. B 594(1–2), 272–286 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Ryu S., Takayanagi T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  35. Ryu S., Takayanagi T.: Aspects of holographic entanglement entropy. JHEP 0608, 045 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  36. Schwimmer A., Theisen S.: Entanglement Entropy, Trace Anomalies and Holography. Nucl. Phys. B 801, 1–24 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  37. Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–326 (1993)

    MATH  MathSciNet  Google Scholar 

  38. Tromba, A.: Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in \({\mathbb{R}^n}\). Part I No. 4(Trans. Am. Math. Soc. 290), 385–413 (1985)

  39. Tonegawa Y.: Existence and regularity of constant mean curvature hypersurfaces in hyperbolic space. Math. Z. 221, 591–615 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  40. Wang, X.: Holography and the geometry of certain convex cocompact hyperbolic 3-manifolds. http://arXiv.org/abs/math/0210164v2[math.DG], 2003

  41. White B.: Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals. Invent. Math. 88, 243–256 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. White B.: The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Ind. Univ. Math. J. 36(3), 567–602 (1987)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Bahen Centre, 40 st. George Street, Toronto, Ontario, M5S2E4, Canada

    Spyridon Alexakis

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Rafe Mazzeo

Authors
  1. Spyridon Alexakis
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  2. Rafe Mazzeo
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Correspondence to Rafe Mazzeo.

Additional information

Communicated by P.T. Chruściel

This research was partially conducted during the period the author served as a Clay Research Fellow.

Supported by the NSF under Grant DMS-0505709.

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Alexakis, S., Mazzeo, R. Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds. Commun. Math. Phys. 297, 621–651 (2010). https://doi.org/10.1007/s00220-010-1054-3

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  • DOI: https://doi.org/10.1007/s00220-010-1054-3

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