Advertisement

Perelman’s Functionals on Cones

Construction of Type III Ricci Flows Coming Out of Cones
  • Tristan Ozuch
Article
  • 11 Downloads

Abstract

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow. In a first part, we study Perelman’s \(\lambda \) and \(\nu \) functionals of cones and characterize their finiteness in terms of the \(\lambda \)-functional of the link. As an application, we characterize manifolds with conical singularities on which a \(\lambda \)-functional can be defined and get upper bounds on the \(\nu \)-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman’s pseudolocality theorem and prove that cones over some perturbations of the unit sphere can be smoothed out by type III immortal solutions of the Ricci flow.

Keywords

Ricci flow Conical singularity Asymptotic cone log-Sobolev inequality Type III solution Perelman’s functionals 

Notes

Acknowledgements

I would like to thank Richard Bamler for inviting me at UC Berkeley and supervising this work.

References

  1. 1.
    Bamler, R., Maximo, D.: Almost-rigidity and the extinction time of positively curved Ricci flows. Math. Ann. 369, 899–911 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beesack, P.: Hardy’s inequality and its extensions. Pac. J. Math. 11(1), 39–61 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brendle, S., Schoen, R.M.: Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc. 22(1), 287–307 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cavaletti, F., Mondino, A.: Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geom. Topol. 21, 603–645 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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. Differ. Geom. 17(1), 15–53 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, I., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: I, II, III and IV. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007)zbMATHGoogle Scholar
  7. 7.
    Dai, X., Wang, C.: Perelman’s \(\lambda \)-functional on manifolds with conical singularities, preprint (2017)Google Scholar
  8. 8.
    Deruelle, A.: Géométrie à l’infini de certaines variétés riemanniennes non compactes (in French), PhD thesis (2012)Google Scholar
  9. 9.
    Deruelle, A.: Smoothing out positively curved metric cones by Ricci expanders. Geom. Funct. Anal. 26(1), 188–249 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gianniotis, P., Schulze, F.: Ricci flow from spaces with isolated conical singularities, preprintGoogle Scholar
  12. 12.
    Hamilton, R.: A compactness property for solutions of the Ricci flow. Am. J. Math. 117(3), 545–572 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Huisken, G.: Ricci deformation of the metric on a Riemannian manifold. J. Differ. Geom. 21(1), 47–62 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12, 2587–2858 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lott, J., Zhang, Z.: Ricci flow on quasiprojective manifolds II. Duke Math. J. 156(1), 87–123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, L.: Ricci expanders and type III Ricci flow, preprintGoogle Scholar
  17. 17.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, preprint (2002)Google Scholar
  18. 18.
    Schulze, F., Simon, M.: Expanding solitons with nonnegative curvature operators coming out of cones. Math. Z. 275(1–2), 625–639 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Simon, M.: Deformation of \(C^0\) Riemannian metrics in the direction of their Ricci curvature. Commun. Anal. Geom. 10(5), 1033–1074 (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Département de Mathématiques et Applications, École Normale Supérieure, PSL Research UniversityParisFrance

Personalised recommendations