Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 15 (2019), 057, 21 pages      arXiv:1607.08558      https://doi.org/10.3842/SIGMA.2019.057

Ricci Flow and Volume Renormalizability

Eric Bahuaud a, Rafe Mazzeo b and Eric Woolgar c
a) Department of Mathematics, Seattle University, 901 12th Ave, Seattle, WA 98122, USA
b) Department of Mathematics, Stanford University, Stanford, CA 94305, USA
c) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

Received December 06, 2018, in final form July 30, 2019; Published online August 07, 2019

Abstract
With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula\[\frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \int_{M^n} (S(g(t))+n(n-1) ) {\rm d}V_{g(t)},\]where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

Key words: Ricci flow; conformally compact metrics; asymptotically hyperbolic metrics; renormalized volume.

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References

  1. Albin P., Renormalizing curvature integrals on Poincaré-Einstein manifolds, Adv. Math. 221 (2009), 140-169, arXiv:math.DG/0504161.
  2. Ammar M., Polyhomogénéité des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci, Ph.D. Thesis, Université du Québec à Montréal, 2019, arXiv:1907.03917.
  3. Bahuaud E., Ricci flow of conformally compact metrics, Ann. Inst. H. Poincar'e Anal. Non Linéaire 28 (2011), 813-835, arXiv:1011.2999.
  4. Bahuaud E., Mazzeo R., Woolgar E., Renormalized volume and the evolution of APEs, Geom. Flows 1 (2015), 126-138, arXiv:1307.4788.
  5. Bahuaud E., Woolgar E., Asymptotically hyperbolic normalized Ricci flow and rotational symmetry, Comm. Anal. Geom. 26 (2018), 1009-1045, arXiv:1506.06806.
  6. Balehowsky T., Woolgar E., The Ricci flow of asymptotically hyperbolic mass and applications, J. Math. Phys. 53 (2012), 072501, 15 pages, arXiv:1110.0765.
  7. Bamler R.H., Stability of symmetric spaces of noncompact type under Ricci flow, Geom. Funct. Anal. 25 (2015), 342-416, arXiv:1011.4267.
  8. Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-Verlag, Berlin, 1987.
  9. Chen B.-L., Zhu X.-P., Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006), 119-154, arXiv:math.DG/0505447.
  10. Chruściel P.T., Delay E., Lee J.M., Skinner D.N., Boundary regularity of conformally compact Einstein metrics, J. Differential Geom. 69 (2005), 111-136, arXiv:math.DG/0401386.
  11. Djadli Z., Guillarmou C., Herzlich M., Opérateurs géométriques, invariants conformes et variétés asymptotiquement hyperboliques, Panoramas et Synthèses, Vol. 26, Société Mathématique de France, Paris, 2008.
  12. Fefferman C., Graham C.R., The ambient metric, Annals of Mathematics Studies, Vol. 178, Princeton University Press, Princeton, NJ, 2012.
  13. Graham C.R., Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo (2) Suppl. (2000), 31-42, arXiv:math.DG/9909042.
  14. Graham C.R., Lee J.M., Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186-225.
  15. Guillarmou C., Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J. 129 (2005), 1-37, arXiv:math.SP/0311424.
  16. Mazzeo R.R., Melrose R.B., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260-310.
  17. Qing J., Shi Y., Wu J., Normalized Ricci flows and conformally compact Einstein metrics, Calc. Var. Partial Differential Equations 46 (2013), 183-211, arXiv:1106.0372.
  18. Rochon F., Polyhomogénéité des métriques asymptotiquement hyperboliques complexes le long du flot de Ricci, J. Geom. Anal. 25 (2015), 2103-2132, arXiv:1305.5457.
  19. Schnürer O.C., Schulze F., Simon M., Stability of hyperbolic space under Ricci flow, Comm. Anal. Geom. 19 (2011), 1023-1047, arXiv:1003.2107.
  20. Shi W.-X., Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), 303-394.
  21. Vasy A., Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. Math. 194 (2013), 381-513, arXiv:1012.4391.

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