Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds

Advances in Mathematics

Volumn 179, Issue 2, 2003, Pages 205-249

  Anderson, Michael T ^a  

^a STONY BROOK UNIVERSITY   (United States)

Author keywords

AdS CFT correspondence; Conformally compact; Einstein metrics

Indexed keywords

EID: 0242595903     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0001-8708(02)00075-0     Document Type: Article

References (39)

1. R.A. Adams, Sobolev spaces, Pure Appl. Math. Ser. 65 (1975).
2. M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calculus Variations Partial Differential Equation 5 (1997) 199-269.
3. 2 curvature and volume renormalization for AHE metrics on 4-manifolds, Math. Res. Lett. 8 (2001) 171-188 math-DG/0011051.
4. M. Anderson, Einstein metrics with prescribed conformal infinity on 4-manifolds, preprint, May 2001, math. DG/0105243.
5. M. Berger, Quelques formules de variation pour une structure Riemannienne, Ann. Sci. Ecole Norm. Sup. 3 (1970) 285-294.
6. A. Besse, Einstein Manifolds, in: Ergebnisse Series, Vol. 3(10), Springer, New York, 1987.
7. O. Biquard, Métriques d'Einstein asymptotiquement symétriques, Astérisque 265 (2000).
8. O. Biquard, Einstein deformations of hyperbolic metrics, in: C. LeBrun, M. Wang (Eds.), Essays on Einstein Manifolds, Surveys in Differential Geometry, Vol. VI, International Press, Cambridge, MA, 1999, pp. 235-246.
9. M. Cai, G.J. Galloway, Boundaries of zero scalar curvature in the AdS/CFT correspondence, Adv. Theoret. Math. Phys. 3 (1999) 1769-1783 hep-th/0003046.
10. A.P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958) 16-36.
11. A.P. Calderón, Existence and uniqueness theorems for systems of partial differential equations, in: Proceedings of the Symposium on Fluid Dynamics and Applied Mathematics, University of Maryland, 1961, Gordon and Breach, New York, 1962, pp. 147-195.
12. J. Cheeger, M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II, J. Differential Geometry 23 (1986) 309-346 32 (1990) 269-298.
13. C.B. Croke, B. Kleiner, A warped product splitting theorem, Duke Math. J. 67 (1992) 571-574.
14. S. de Haro, K. Skenderis, S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217 (2001) 595-622 hep-th/0002230.
15. J. Escobar, On the prescribed scalar curvature problem on compact manifolds with boundary, Contemp. Math. 268 (2000) 137-144.
16. C. Fefferman, C.R. Graham, Conformal invariants, in: Élie Cartan et les Mathematiques d'Aujourd'hui, Astérisque, 1985, Numero Hors Serie, Soc. Math. France, Paris, pp. 95-116.
17. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer, New York, 1983.
18. C.R. Graham, Volume and area normalization for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo Ser. II 63 (Suppl.) (2000) 31-42 math.DG/0009042.
19. C.R. Graham, J.M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. in Math. 87 (1991) 186-225.
20. M. Gromov, Hyperbolic manifolds according to Thurston and Jorgensen, Sem. Bourbaki 546 (1980), Lecture Notes in Math., Vol. 842, Springer Verlag, Berlin, 1981, 40-53.
21. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Prog. Math. Ser. 152 (1999).
22. M. Gursky, Compactness of conformal metrics with integral bounds on curvature, Duke Math. J. 72 (1993) 339-367.
23. S.W. Hawking, C.J. Hunter, D.N. Page, Nut charge, Anti-de Sitter space and entropy, Phys. Rev. D 59 (1999) 044033 hep-th/9809035.
24. S.W. Hawking, D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Comm. Math. Phys. 87 (1983) 577-588.
25. M. Henningson, K. Skenderis, The holographic Weyl anomaly, J. High Energy Phys. 9807 (1998) 023 hep-th/9806087.
26. A. Kasue, Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan 35 (1983) 117-131.
27. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience, Wiley, New York, 1963.
28. C. LeBrun, ℋ-space with a cosmological constant, Proc. Royal Soc. London Ser. A 380 (1982) 171-185.
29. M. Li, The Yamabe problem with Dirichlet data, C. R. Acad. Sci. Paris Sér. I 320 (1995) 709-712.
30. J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, I, in: Grundlehren Series, Vol. 181, Springer, New York, 1972.
31. J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theoret. Math. Phys. 2 (1998) 231 252 hep-th/9711200.
32. C.B. Morrey Jr., Multiple Integrals in the Calculus of Variations, in: Grundlehren Series, Vol. 130, Springer, Berlin, 1966.
33. R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, (preprint): hep-th/9903203.
34. L. Nirenberg, "Lectures on Partial Differential Equations", in: CMBS-NSF Regional Conference Series in Mathematics, Vol. 17, Amer. Math. Soc, Providence, RI, 1973.
35. P. Petersen, "Riemannian Geometry", in: Graduate Texts in Mathematics, Vol. 171, Springer, New York, 1997.
36. K.T. Smith, Some remarks on a paper of Calderon on existence and uniqueness theorems for systems of partial differential equations, Comm. Pure Appl. Math. 18 (1965) 415-441.
37. W. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University, Princeton, NJ, 1978, preprint.
38. E. Witten, Anti De Sitter space and holography, Adv. Theoret. Math. Phys. 2 (1998) 253-291 hep-th/9802150.
39. E. Witten, S.-T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theoret. Math. Phys. 3 (1999) 1635-1655 hep-th/9910245.