Abstract
We prove precompactness in an orbifold Cheeger–Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss–Bonnet with cutoff argument.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abresch U., Gromoll D.: On complete manifolds with nonnegative Ricci curvature. J. Amer. Math. Soc. 3(2), 355–374 (1990)
Anderson M.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 23, 455–490 (1989)
Anderson M., Cheeger J.: Diffeomorphism finiteness for manifolds with Ricci curvature and L n/2-norm of curvature bounded, Geom. Funct. Anal. 1(3), 231–252 (1991)
Bando S., Kasue A., Nakajima H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97(2), 313–349 (1989)
L. Bessières, G. Besson, M. Boileau, S. Maillot, J. Porti, Geometrisation of 3-Manifolds, EMS Tracts in Mathematics 13, Zürich (2010).
H.-D. Cao, Recent progress on Ricci solitons, in “Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM) 11, International Press, (2009).
Cao H.-D., Sesum N.: A compactness result for Kähler Ricci solitons. Adv. Math. 211(2), 794–818 (2007)
Cao H.-D., Zhou D.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85(2), 175–186 (2010)
H.-D. Cao, X.-P. Zhu, Hamilton-Perelman’s proof of the Poincaré conjecture and the geometrization conjecture, (2006); arXiv:math/0612069v1
Carrillo J., Ni L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. Comm. Anal. Geom. 17(4), 721–753 (2009)
Cheeger J.: Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92, 61–74 (1970)
Cheeger J.: Degeneration of Einstein metrics and metrics with special holonomy. Surveys Diff. Geom. VIII: 29(−73), 29–73 (2003)
Cheeger J., Gromov M., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Diff. Geom. 17(1), 15–53 (1982)
X. Chen, B. Wang, Space of Ricci flows (I), preprint (2009); arXiv:0902.1545v1
B. Chow, D. Knopf, The Ricci Flow: An Introduction, Mathematical Surveys and Monographs 110, AMS, Providence (2004).
Croke C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. (4) 13(4), 419–435 (1980)
Eguchi T., Hanson A.: Gravitational instantons. Gen. Relativity Gravitation 11(5), 315–320 (1979)
Feldman M., Ilmanen T., Knopf D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Diff. Geom. 65(2), 169–20 (2003)
P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Math. Lect. Series 11, Publish or Perish (1984).
Greene R., Wu H.: Lipschitz convergence of Riemannian manifolds. Pacific J. Math. 131(1), 119–141 (1988)
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkhäuser Boston (1999).
Hamilton R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17(2), 255–306 (1982)
Hamilton R.: A compactness property for solutions of the Ricci flow. Amer. J. Math. 117, 545–572 (1995)
Hamilton R.: The formation of singularities in the Ricci flow. Surveys Diff. Geom. II: 7, 7–136 (1995)
Kleiner B., Lott J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)
J. Morgan, G. Tian, Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3, AMS, Cambridge, MA (2007).
O. Munteanu, The volume growth of complete gradient shrinking Ricci solitons, preprint (2009); arXiv:0904.0798v2
O. Munteanu, N. Sesum, On gradient Ricci solitons, preprint (2009); arXiv:0910.1105v1
Nakajima H.: Hausdorff convergence of Einstein 4-manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35(2), 411–424 (1988)
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002); arXiv:math/0211159v1
Sibner L.: The isolated point singularity problem for the coupled Yang–Mills equations in higher dimensions. Math. Ann. 271(1), 125–131 (1985)
J. Song, B. Weinkove, Contracting exceptional divisors by the Kähler-Ricci flow, preprint (2010); arXiv:1003.0718v1
Tian G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101(1), 101–172 (1990)
Tian G., Viaclovsky J.: Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160(2), 357–415 (2005)
Tian G., Viaclovsky J.: Moduli spaces of critical Riemannian metrics in dimension four. Adv. Math. 196(2), 346–372 (2005)
Tian G., Viaclovsky J.: Volume growth, curvature decay, and critical metrics. Comment. Math. Helv. 83(4), 889–911 (2008)
Uhlenbeck K.: Removable singularities in Yang–Mills fields. Comm. Math. Phys. 83(1), 11–29 (1982)
C. Villani, Optimal transport. Old and new, Grundlehren der Mathematischen Wissenschaften 338, Springer-Verlag (2009).
Weber B.: Convergence of compact Ricci Solitons. Int. Math. Res. Not. 1, 96–118 (2011)
Wei G., Wylie W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Diff. Geom. 83(2), 337–405 (2009)
Zhang X.: Compactness theorems for gradient Ricci solitons. J. Geom. Phys. 56(12), 2481–2499 (2006)
Zhang Z.H.: On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc. 137, 8–27552759 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Haslhofer, R., Müller, R. A Compactness Theorem for Complete Ricci Shrinkers. Geom. Funct. Anal. 21, 1091–1116 (2011). https://doi.org/10.1007/s00039-011-0137-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-011-0137-4