The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1157–1174 | Cite as

On Moduli Spaces of Ricci Solitons

  • Fabio PodestàEmail author
  • Andrea Spiro


We study deformations of shrinking Ricci solitons on a compact manifold M, generalizing the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pre-moduli spaces. We prove the existence of a finite-dimensional submanifold of \(S_{f}\times\mathcal{C}^{\infty}(M)\), which contains the pre-moduli space of solitons around a fixed shrinking Ricci soliton as an analytic subset. We define solitonic rigidity and give criteria which imply it.


Ricci soliton Ebin slice Space of Riemannian metrics 

Mathematics Subject Classification

53C25 53C21 


  1. 1.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987) CrossRefzbMATHGoogle Scholar
  2. 2.
    Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. 167, 1079–1097 (2008) CrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, H.-D.: Geometry of Ricci solitons. Chin. Ann. Math., Ser. B 27, 121–142 (2006) CrossRefzbMATHGoogle Scholar
  5. 5.
    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 I. Geometric Aspects. Am. Math. Soc., Providence (2007) Google Scholar
  6. 6.
    Cahn, R.S., Wolf, J.A.: Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one. Comment. Math. Helv. 51, 1–21 (1976) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cao, H.-D., Zhu, M.: On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353, 747–763 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dancer, A.S., Wang, M.Y.: On Ricci solitons of cohomogeneity one. Ann. Glob. Anal. Geom. 39, 259–292 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ebin, D.G.: On the space of Riemannian metrics. Doctoral Thesis, Massachusetts Institute of Technology, Cambridge (1967) Google Scholar
  10. 10.
    Ebin, D.G.: The manifolds of Riemannian metrics. In: Proceedings of the Symposia on Pure Mathematics, vol. XV (1970) Google Scholar
  11. 11.
    Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons—the equation point of view. Manuscr. Math. 127, 345–367 (2008) CrossRefzbMATHGoogle Scholar
  12. 12.
    Futaki, A., Li, H., Li, X.-D.: On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons (2012). arXiv:1111.6364v4
  13. 13.
    Koiso, N.: Rigidity and stability of Einstein metrics—the case of compact symmetric spaces. Osaka J. Math. 17, 51–73 (1980) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Koiso, N.: Rigidity and infinitesimal deformability of Einstein metrics. Osaka J. Math. 19, 643–668 (1982) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73, 71–106 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Springer, Berlin (2005) zbMATHGoogle Scholar
  17. 17.
    Hall, S.J., Murphy, T.: On the linear stability of Kähler-Ricci solitons. Proc. Am. Math. Soc. 139(9), 3327–3337 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Li, H.: Complex deformation of critical Kähler metrics (2012). arXiv:1206.0912
  19. 19.
    Palais, R.S.: On the differentiability of isometries. Proc. Am. Math. Soc. 8, 805–807 (1957) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Palais, R.S.: In: Atiyah, M.F., Borel, A., Floyd, E.E., Seeley, R.T., Shih, W., Solovay, R. (eds.): Seminar on the Atiyah-Singer Theorem. Princeton University Press, Princeton (1965) Google Scholar
  21. 21.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric application (2008). arXiv:0801.3504v1
  22. 22.
    Podestà, F., Spiro, A.: Kähler-Ricci solitons on homogeneous toric bundles. J. Reine Angew. Math. 642, 109–127 (2010) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Simon, U.: Curvature bounds for the spectrum of closed Einstein spaces. Can. J. Math. 30, 1087–1091 (1978) CrossRefzbMATHGoogle Scholar
  24. 24.
    Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101, 101–172 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tian, G., Zhu, X.: A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment. Math. Helv. 77, 297–325 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Topping, P.: Lectures on the Ricci Flow. Cambridge University Press, Cambridge (2006) CrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, X.-J., Zhu, X.: Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Dip. di Matematica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.Scuola di Scienze e TecnologieUniversità di CamerinoCamerino (Macerata)Italy

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