# Convergence Stability for Ricci Flow

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## Abstract

The principle of *convergence stability* for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state \(g_0\) exists for all time and converges to a stable fixed point, then the flows of solutions that start near \(g_0\) also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries.

## Keywords

Stability of the Ricci flow Continuous dependence of the Ricci flow Analytic semigroups## Mathematics Subject Classification

53C44 58J35 35K## Notes

### Acknowledgements

The authors would like to thank Dan Knopf, Jack Lee, Rafe Mazzeo, and Haotian Wu for helpful conversations related to this work. We are grateful to the anonymous referee who made several suggestions that improved this paper. This work was supported by Grants from the Simons Foundation (#426628, E. Bahuaud and #283083, C. Guenther). J. Isenberg was partially supported by the NSF Grant DMS-1263431. This work was initiated at the 2015 BIRS workshop *Geometric Flows: Recent Developments and Applications* (15w5148).

## References

- 1.Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math.
**17**, 35–92 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Amann, H.: Linear and quasilinear parabolic problems. Vol. I. Abstract linear theory. Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, (1995)Google Scholar
- 3.Angenent, S., Daskalopoulos, P., Sesum, N.: Unique asymptotics of ancient convex mean curvature flow solutions, arXiv:1503.01178
- 4.Bamler, R.: Stability of hyperbolic manifolds with cusps under Ricci flow. Adv. Math.
**263**, 412–467 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bamler, R.: Stability of symmetric spaces of noncompact type under Ricci flow. Geom. Funct. Anal.
**25**(2), 342–416 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Bamler, R., Kleiner, B.: Uniqueness and stability of Ricci flow through singularities, arXiv:1709.04122
- 7.Besse, A.L.: Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, (2008). xii+516 ppGoogle Scholar
- 8.Biquard, O., Mazzeo, R.: A nonlinear Poisson transform for Einstein metrics on product spaces. J. Eur. Math. Soc.
**13**(5), 1423–1475 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Chow, B., Knopf, D.: The Ricci Flow: An Introduction. American Mathematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
- 10.Chow, B., Chu, S-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: techniques and applications. Part IV. Long-time solutions and related topics. Math. Surv. Mono., 206. AMS, Providence, RI, pp. xx+374 (2015)Google Scholar
- 11.Chow, B., Lu, P., Lei, N.: Hamilton’s Ricci flow. Grad. Stud. Math.
**77**, 608 (2006)MathSciNetzbMATHGoogle Scholar - 12.Carfora, M., Isenberg, J., Jackson, M.: Convergence of the Ricci flow for metrics with indefinite Ricci curvature. J. Differ. Geom.
**31**(1), 249–263 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Clément, P.: One-parameter semigroups. CWI Monographs, North-Holland (1987)zbMATHGoogle Scholar
- 14.Dai, X., Wang, X., Wei, G.: On the variational stability of Kaehler–Einstein metrics. Commun. Anal. Geom.
**15**(4), 669–693 (2007)CrossRefzbMATHGoogle Scholar - 15.DaPrato, G., Lunardi, A.: Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach Space. Arch. Ration. Mech. Anal.
**101**, 115–141 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Eichhorn, J.: Global Analysis on Open Manifolds. Nova Science, New York (2007)zbMATHGoogle Scholar
- 17.Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, pp. xiv+517 (2001)Google Scholar
- 18.Guenther, C., Isenberg, J., Knopf, D.: Stability of the Ricci flow at Ricci-flat metrics. Commun. Anal. Geom.
**10**(4), 741–777 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom.
**17**, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Haslhofer, R., Muller, R.: Dynamical stability and instability of Ricci-flat metrics. Math. Ann.
**360**, 547–553 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Isenberg, J., Jackson, M.: Ricci flow of locally homogeneous geometries on closed manifolds. J. Differ. Geom.
**35**(3), 723–741 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Jost, J.: Riemannian geometry and geometric analysis. Universitext, Springer, New York (2011)CrossRefzbMATHGoogle Scholar
- 23.Knopf, D., Young, A.: Asymptotic stability of the cross curvature flow at a hyperbolic metric. Proc. Am. Math. Soc.
**137**, 699–709 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Koch, H., Lamm, T.: Geometric flows with rough initial data. Asian J. Math.
**16**(2), 209–235 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Lott, J., Sesum, N.: Ricci flow on three-dimensional manifolds with symmetry. Commun. Math. Helv.
**89**(1), 1–32 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Li, H., Yin, H.: On stability of the hyperbolic space form under the normalized Ricci flow. Int. Math. Res. Not.
**5**, 2903–2924 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Lorenzi, L., Lunardi, A., Metafune, G., Pallara, D.: Analytic semigroups and reaction-diffusion problems. Internet seminar, 2004–2005Google Scholar
- 28.Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. Progress in nonlinear differential equations and their applications 16, Birkhäuser Boston, Boston, MA, (1995)Google Scholar
- 29.MathOverFlow question by Igor Belegradek,
*Does Ricci flow depend continuously on the initial metric*, https://mathoverflow.net/questions/58480 - 30.Chow, B., Chu, S., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D. Lu, P., Luo, F., Ni, L.: The Ricci flow: Techniques and Applications. Part IV: Long Time Solutions and Related Topics. Mathematical Surveys and Monographs, 206, AMS, Providence, RI (2015)Google Scholar
- 31.Sesum, N.: Linear and dynamical stability of Ricci flat metrics. Duke Math. J.
**133**(1), 1–26 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Schnurer, O., Schultze, F., Simon, M.: Stability of Euclidean space under Ricci flow Comm. Anal. Geom.
**16**(1), 127–158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Schnurer, O., Schultze, F., Simon, M.: Stability of hyperbolic space under Ricci flow Comm. Anal. Geom.
**19**(5), 1023–1047 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Wu, H.: Stability of complex hyperbolic space under curvature-normalized Ricci flow Geom. Dedicata
**164**(1), 231–258 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 35.Williams, M., Wu, H.: Dynamical stability of algebraic Ricci solitons. J. Reine Angew. Math.
**713**, 225–243 (2016)MathSciNetzbMATHGoogle Scholar