Skip to main content Accessibility help
×
Home

THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES

  • Colin Guillarmou (a1), Sergiu Moroianu (a2) and Jean-Marc Schlenker (a3)

Abstract

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$ , we denote it by $\text{Vol}_{R}(M,g;h_{0})$ . We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$ , satisfied in particular by Einstein metrics. When $n=2$ , choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$ -Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$ , the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$ . We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

Copyright

Footnotes

Hide All

C. G. was partially supported by the A.N.R. project ACG ANR-10-BLAN-0105. S. M. was partially supported by the CNCS project PN-II-RU-TE-2011-3-0053; he thanks the Fondation des Sciences Mathématiques de Paris and the École Normale Supérieure for additional support. J.-M. S. was partially supported by the A.N.R. through projects ETTT, ANR-09-BLAN-0116-01, and ACG, ANR-10-BLAN-0105.

Footnotes

References

Hide All
1. Ahlfors, L. and Bers, L., Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385404.
2. Albin, P., Renormalizing curvature integrals on Poincaré–Einstein manifolds, Adv. Math. 221(1) (2009), 140169.
3. Anderson, M. T., L 2 curvature and renormalization of AHE metrics on 4-manifolds, Math. Res. Lett. 8 (2001), 171188.
4. Bär, C., Gauduchon, P. and Moroianu, A., Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249(3) (2005), 545580.
5. Bers, L., Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 9497.
6. Besse, A., Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 10 (Springer, Berlin, 1987).
7. Biquard, O., Métriques d’Einstein asymptotiquement symétriques, Astérisque 265 (2000).
8. Biquard, O., Autodual Einstein versus Kähler–Einstein, Geom. Funct. Anal. 15(3) (2005), 598633.
9. Biquard, O., Continuation unique à partir de l’infini conforme pour les métriques d’Einstein, Math. Res. Lett. 15(6) (2008), 10911099.
10. Bochner, S., Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776797.
11. Caffarelli, L., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155(3–4) (1985), 261301.
12. Chang, S.-Y. A. and Fang, H., A class of variational functionals in conformal geometry, Int. Math. Res. Not. IMRN (7) (2008), Art. ID rnn008.
13. Chang, S.-Y. A., Fang, H. and Graham, C. R., A note on renormalized volume functionals, Differential Geom. Appl. 33(Suppl) (2014), 246258.
14. Chang, S.-Y. A., Gursky, M. J. and Yang, P. C., An equation of Monge–Ampre type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155(3) (2002), 709787.
15. Chang, S.-Y. A., Gursky, M. J. and Yang, P. C., An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002), 151186. Dedicated to the memory of Thomas H. Wolff.
16. Chang, S.-Y. A., Qing, J. and Yang, P., On the topology of conformally compact Einstein 4-manifolds, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemporary Mathematics, Volume 350, pp. 4961 (American Mathematical Society, Providence, RI, 2004).
17. Chang, S.-Y. A., Qing, J. and Yang, P. C., On the renormalized volumes for conformally compact Einstein manifolds, in Proceeding Conf. in Geometric Analysis, Moscow 2005.
18. Chrusciel, P., Delay, E., Lee, J. M. and Skinner, D. N., Boundary regularity of conformally compact Einstein metrics, J. Differential Geom. 69(1) (2005), 111136.
19. de Haro, S., Skenderis, K. and Solodukhin, S. N., Holographic reconstruction of space time and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217 (2001), 595622.
20. Delay, E., TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics, Manuscripta Math. 123(2) (2007), 147165.
21. Ebin, D. G., The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), pp. 1140 (American Mathematical Society, Providence, RI, 1970).
22. Djadli, Z., Guillarmou, C. and Herzlich, M., Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques, Panoramas et Synthèses, Volume 26 (Société Mathématique de France, Paris, 2008).
23. Fefferman, C. and Graham, C. R., The Ambient Metric, Annals of Mathematics Studies, Volume 178 (Princeton University Press, Princeton, NJ, 2012), x+113 pp.
24. Fischer, A. E. and Moncrief, V., The structure of quantum conformal superspace, in Global Structure and Evolution in General Relativity (Karlovassi, 1994), Lecture Notes in Physics, Volume 460, pp. 111173 (Springer, Berlin, 1996).
25. Frankel, T., On theorems of Hurwitz and Bochner, J. Math. Mech. 15 (1966), 373377.
26. Graham, C. R., Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo Ser. II 63(Suppl) (2000), 3142.
27. Graham, C. R., Dirichlet-to-Neumann map for Poincaré–Einstein metrics, Announc. Oberwolfach Rep. 2(3) (2005), 22002203.
28. Graham, C. R., Extended obstruction tensors and renormalized volume coefficients, Adv. Math. 220(6) (2009), 19561985.
29. Graham, C. R. and Hirachi, K., The ambient obstruction tensor and Q-curvature, in AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lectures in Mathematics and Theoretical Physics, Volume 8, pp. 5971 (European Mathematical Society, Zürich, 2005).
30. Graham, C. R. and Juhl, A., Holographic formula for Q-curvature, Adv. Math. 216(2) (2007), 841853.
31. Graham, C. R. and Lee, J., Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186225.
32. Graham, C. R. and Zworski, M., Scattering matrix in conformal geometry, Invent. Math. 152(1) (2003), 89118.
33. Guillarmou, C., Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129(1) (2005), 137.
34. Guillarmou, C. and Moroianu, S., Chern-Simons line bundle on Teichmüller space, Geom. Topol. 18(1) (2014), 327377.
35. Guan, P., Viaclovsky, J. and Wang, G., Some properties of the Schouten tensor and applications to conformal geometry, Trans. Amer. Math. Soc. 355 (2003), 925933.
36. Guillopé, L. and Zworski, M., Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal. 129(2) (1995), 363389.
37. Gursky, M. and Viaclovsky, J., Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J. 52(2) (2003), 399419.
38. Gursky, M. and Viaclovsky, J., Volume comparison and the k-Yamabe problem, Adv. Math. 187(2) (2004), 447487.
39. Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7(1) (1982), 65222.
40. Henningson, M. and Skenderis, K., The holographic Weyl anomaly, J. High Energy Phys. (7) (1998), Paper 23, 12 pp. (electronic).
41. Juhl, A., Families of Conformally Covariant Differential Operators, Q-Curvature and Holography, Progress in Mathematics, Volume 275 (Birkhäuser, 2009).
42. Krasnov, K., Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4(4) (2000), 929979.
43. Krasnov, K. and Schlenker, J.-M., On the renormalized volume of hyperbolic 3-manifolds, Comm. Math. Phys. 279(3) (2008), 637668.
44. Lee, J. M., Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183(864) (2006).
45. Marden, A., The geometry of finitely generated kleinian groups, Ann. of Math. (2) 99 (1974), 383462.
46. Marden, A., Deformation of Kleinian groups, in Handbook of Teichmüller Theory, Volume I,(ed. Papadopoulos, A.), IRMA Lectures in Mathematics and Theoretical Physics, Volume 11 (European Mathematical Society, 2007). chapter 9.
47. Matsumoto, Y., A GJMS construction for 2-tensors and the second variation of the total Q-curvature, Pacific J. Math. 262(2) (2013), 437455.
48. Mazzeo, R., Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16(10) (1991), 16151664.
49. Melrose, R. B., Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Not. 3 (1992), 5161.
50. Melrose, R. B., Manifolds with Corners, Available at http://math.mit.edu/∼rbm/book.html (in preparation).
51. Mcmullen, C. T., The moduli space of Riemann surfaces is Kähler hyperbolic, Ann. of Math. (2) 151(1) (2000), 327357.
52. Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333340.
53. Obata, M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247258.
54. Osgood, B., Phillips, R. and Sarnak, P., Extremals for determinants of Laplacians, J. Funct. Anal. 80 (1988), 148211.
55. Patterson, S. J. and Perry, P., The divisor of Selberg’s zeta function for Kleinian groups (appendix A by Charles Epstein), Duke Math. J. 106 (2001), 321391.
56. Payne, K. R., Smooth tame Fréchet algebras and Lie groups of pseudodifferential operators, Comm. Pure Appl. Math. 44(3) (1991), 309337.
57. Reilly, R. C., On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373383.
58. Rivin, I. and Schlenker, J.-M., The Schläfli formula in Einstein manifolds with boundary, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 1823.
59. Schlenker, J.-M., The renormalized volume and the volume of the convex core of quasifuchsian manifolds, Math. Res. Lett. 20(4) (2013), 773786.
60. Sheng, W., Trudinger, N. S. and Wang, X.-J., The Yamabe problem for higher order curvature, J. Differential Geom. 77 (2007), 515553.
61. Skenderis, K. and Solodukin, S. N., Quantum effective action from the AdS/CFT correspondence, Phys. Lett. B 472 (2000), 316322.
62. Takhtajan, L. A. and Teo, L.-P., Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239(1–2) (2003), 183240.
63. Taylor, M. E., Partial differential equations III, in Nonlinear Equations, 2nd ed., Applied Mathematical Sciences, Volume 117 (Springer, New York, 2011), xxii+715 pp.
64. Viaclosvky, J., Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom. 10(4) (2002), 815846.
65. Wang, F., Dirichlet-to-Neumann map for Poincaré–Einstein metrics in even dimension, arXiv:0905.2457.
66. Yano, K., On Harmonic and Killing Vector Fields, Ann. of Math. (2) 55(1) (1952), 3845.
67. Zograf, P. G., Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces, Algebra i Analiz 1(4) (1989), 136160; (in Russian), Engl. transl.: Leningrad Math. J. 1(4) (1990), 941–965.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES

  • Colin Guillarmou (a1), Sergiu Moroianu (a2) and Jean-Marc Schlenker (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.