Ricci flow of conformally compact metrics
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 6, p. 813-835

In this paper we prove that given a smoothly conformally compact asymptotically hyperbolic metric there is a short-time solution to the Ricci flow that remains smoothly conformally compact and asymptotically hyperbolic. We adapt recent results of Schnürer, Schulze and Simon to prove a stability result for conformally compact Einstein metrics sufficiently close to the hyperbolic metric.

Lʼobjectif de cet article est de démontrer lʼexistence dʼune solution en temps court du flot de Ricci dans la classe de métriques régulières, conformément compactes et asymptotiquement hyperboliques. Nous appliquons ensuite les résultats de Schnürer, Schulze et Simon pour prouver la stabilité des métriques dʼEinstein conformément compactes suffisamment proches de la métrique hyperbolique.

DOI : https://doi.org/10.1016/j.anihpc.2011.03.007
Classification:  53C44,  58J35,  35K40,  35K59
Keywords: Ricci flow, Conformally compact metrics, Asymptotically hyperbolic metrics
@article{AIHPC_2011__28_6_813_0,
     author = {Bahuaud, Eric},
     title = {Ricci flow of conformally compact metrics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {6},
     year = {2011},
     pages = {813-835},
     doi = {10.1016/j.anihpc.2011.03.007},
     zbl = {1235.53066},
     mrnumber = {2859929},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_6_813_0}
}
Bahuaud, Eric. Ricci flow of conformally compact metrics. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 6, pp. 813-835. doi : 10.1016/j.anihpc.2011.03.007. http://www.numdam.org/item/AIHPC_2011__28_6_813_0/

[1] Pierre Albin, A renormalized index theorem for some complete asymptotically regular metrics: the Gauss–Bonnet theorem, Adv. Math. 213 no. 1 (2007), 1-52 | MR 2331237 | Zbl 1195.58008

[2] Pierre Albin, Clara L. Aldana, Frédéric Rochon, Ricci flow and the determinant of the Laplacian on non-compact surfaces, 2009.

[3] Eric Bahuaud, Emily Dryden, Boris Vertman, Mapping properties of the heat operator on edge manifolds, 2011.

[4] Eric Bahuaud, Romain Gicquaud, Conformal compactification of asymptotically locally hyperbolic metrics, J. Geom. Anal. (2010), doi:10.1007/s12220-010-9179-3, in press. | MR 2836592

[5] Eric Bahuaud, Dylan Helliwell, Short-time existence for some higher-order geometric flows, Comm. Partial Differential Equations (2011), in press. | MR 2852074

[6] Richard Bamler, Stability of hyperbolic manifolds with cusps under Ricci flow, 2010.

[7] Olivier Biquard (ed.), AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, IRMA Lectures in Mathematics and Theoretical Physics vol. 8, European Mathematical Society (EMS), Zürich (2005) | MR 2160864 | Zbl 1062.81002

[8] Bing-Long Chen, Xi-Ping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 no. 1 (2006), 119-154 | MR 2260930 | Zbl 1104.53032

[9] Bennett Chow, Peng Lu, Lei Ni, Hamiltonʼs Ricci Flow, Graduate Studies in Mathematics vol. 77, American Mathematical Society, Providence, RI (2006) | MR 2274812 | Zbl 1118.53001

[10] C. Robin Graham, John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 no. 2 (1991), 186-225 | MR 1112625 | Zbl 0765.53034

[11] Daniel Grieser, Basics of the b-calculus, Approaches to Singular Analysis, Berlin, 1999, Oper. Theory Adv. Appl. vol. 125, Birkhäuser, Basel (2001), 30-84 | MR 1827170 | Zbl 0987.58011

[12] Xue Hu, Jie Qing, Yuguang Shi, Regularity and rigidity of asymptotically hyperbolic manifolds, 2009.

[13] James Isenberg, Rafe Mazzeo, Natasha Sesum, Ricci flow on asymptotically conical surfaces with nontrivial topology, 2010.

[14] Lizhen Ji, Rafe Mazzeo, Natasa Sesum, Ricci flow on surfaces with cusps, Math. Ann. 345 no. 4 (2009), 819-834 | MR 2545867 | Zbl 1176.53067

[15] Herbert Koch, Tobias Lamm, Geometric flows with rough initial data, 2009.

[16] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics vol. 12, American Mathematical Society, Providence, RI (1996) | MR 1406091 | Zbl 0865.35001

[17] O.A. Ladyženskaja, V.A. Solonnikov, N.N. UralʼCeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs vol. 23, American Mathematical Society, Providence, RI (1967) | MR 241822

[18] John M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold, Comm. Anal. Geom. 3 no. 1–2 (1995), 253-271 | MR 1362652 | Zbl 0934.58029

[19] John M. Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183 no. 864 (2006) | MR 2252687 | Zbl 1112.53002

[20] Haozhao Li, Hao Yin, On stability of the hyperbolic space form under the normalized Ricci flow, Int. Math. Res. Not. IMRN 15 (2010), 2903-2924 | MR 2673714 | Zbl 1214.53054

[21] Li Ma, Xingwang Xu, Ricci flow with hyperbolic warped product metrics, arXiv preprint, 2007. | MR 2663765

[22] Rafe Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 no. 10 (1991), 1615-1664 | MR 1133743 | Zbl 0745.58045

[23] Rafe Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J. 40 no. 4 (1991), 1277-1299 | MR 1142715 | Zbl 0770.53032

[24] Rafe Mazzeo, Frank Pacard, Maskit combinations of Poincaré–Einstein metrics, Adv. Math. 204 no. 2 (2006), 379-412 | MR 2249618 | Zbl 1097.53029

[25] Rafe Mazzeo, Michael Taylor, Curvature and uniformization, Israel J. Math. 130 (2002), 323-346 | MR 1919383 | Zbl 1003.30031

[26] Richard B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics vol. 4, AK Peters Ltd., Wellesley, MA (1993) | MR 1348401 | Zbl 0796.58050

[27] Todd A. Oliynyk, Eric Woolgar, Rotationally symmetric Ricci flow on asymptotically flat manifolds, Comm. Anal. Geom. 15 no. 3 (2007), 535-568 | MR 2379804 | Zbl 1138.53057

[28] Qing, Shi, Wu, Normalized Ricci flow and conformally compact Einstein metrics, preprint, June 2011. | MR 3016507

[29] Oliver C. Schnürer, Felix Schulze, Miles Simon, Stability of hyperbolic space under Ricci flow, 2010.

[30] Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 no. 1 (1989), 223-301 | MR 1001277 | Zbl 0676.53044