Stability of Ricci de Turck flow on singular spaces

  • Klaus KrönckeEmail author
  • Boris Vertman


In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Mathematics Subject Classification

Primary 53C44 Secondary 53C25 58J35 



The second author thanks Jan Swoboda for important discussions about aspects of Ricci flow. Both authors thank the Geometry at Infinity Priority program of the German Research Foundation DFG for its financial support and for providing a platform for joint research. The authors greatfully acknowledge hospitality of the Mathematical Institutes at Hamburg and Oldenburg Universities.


  1. 1.
    Bahuaud, E., Vertman, B.: Yamabe flow on manifolds with edges. Math. Nachr. 287(23), 127–159 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bahuaud, E., Vertman, B..: Long time existence of the edge Yamabe flow, in preparation (2017)Google Scholar
  3. 3.
    Ballmann, W..: Lectures on Kähler manifolds. In: ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS) Zürich (2006)Google Scholar
  4. 4.
    Besse, A.L.: Einstein manifolds. Reprint of the 1987 Edition. Springer, Berlin (2008)Google Scholar
  5. 5.
    Brüning, J., Lesch, M.: Kähler–Hodge theory for conformal complex cones. Geom. Funct. Anal. 3(5), 439–473 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, X., Wang, Y.: Bessel functions, heat kernel and the conical Kähler–Ricci flow. J. Funct. Anal. 269(2), 551–632 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deruelle, A.: Smoothing out positively curved metric cones by Ricci expanders. Geom. Func. Anal. 26, 188–249 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deruelle, A., Kröncke, K.: Stability of ALE Ricci-flat manifolds under Ricci flow, preprint on arXiv:1707.09919 [math.DG] (2017)
  9. 9.
    Donaldson, S.K.: Kähler metrics with cone singularities along a divisor. In: Pardalos, P., Rassias, T. (eds.) Essays in Mathematics and its Applications, pp. 49–79. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Gallot, S.: Équations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. 12(2), 235–267 (1979)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giesen, G., Topping, P.M.: Ricci flow of negatively curved incomplete surfaces. Calc. Var. Partial Differ. Equ. 38(3–4), 357–367 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Giesen, G., Topping, P.M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36(10), 1860–1880 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hamilton, R.S.: Three-orbifolds with positive Ricci curvature. In: Cao, H.D., et al. (eds.) Collected Papers on Ricci Flow. Series Geometry and Topology, vol. 37, pp. 163–165. International Press, Somerville (2003)Google Scholar
  14. 14.
    Hein, H.-J., Sun, S.: Calabi–Yau Manifolds with Isolated Conical Singularities. Publ. Math. IHES 126, 73–130 (2017)Google Scholar
  15. 15.
    Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler-Einstein metrics with Edge Singularities. Ann. Math. 183(1), 95–176 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)Google Scholar
  17. 17.
    Kirsten, K., Loya, P., Park, J.: Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone. Manuscripta Math. 125(1), 95–126 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures. III. Stability theorems for complex structures. Ann. Math. 71(1), 43–76 (1960)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kröncke, K.: Stability of Einstein manifolds. Ph.D. Thesis, Universität Potsdam (2013)Google Scholar
  21. 21.
    Kröncke, K.: Stable and unstable Einstein warped products. Trans. Am. Math. Soc. 369, 6537–6563 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kröncke, K.: Stability of sin-cones and cosh-cylinders. Ann. Sci. Norm. Super. Pisa, Cl. Sci. 18(3), 1155–1187 (2018)Google Scholar
  23. 23.
    Kühnel, W., Rademacher, H.-B.: Conformal diffeomorphisms preserving the Ricci tensor. Proc. Am. Math. Soc. 123(9), 2841–2849 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale. Publications Mathématiques de l’IHÉS 10(1), 5–56 (1961)CrossRefGoogle Scholar
  25. 25.
    Liu, J., Zhang, X.: Conical Kähler-Ricci flows on Fano manifolds. Adv. Math. 307, 1324–1371 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mazzeo, R., Rubinstein, Y., Sesum, N.: Ricci flow on surfaces with conic singularities. Anal. PDE 8(4), 839–882 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mazzeo, R., Vertman, B.: Analytic torsion on manifolds with edges. Adv. Math. 231(2), 1000–1040 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Melrose, R.B.: The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, vol. 4. A K Peters Ltd., Wellesley (1993)CrossRefGoogle Scholar
  30. 30.
    Melrose, R.B.: Calculus of conormal distributions on manifolds with corners. Int. Math. Res. Not. 3, 51–61 (1992)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pacini, T.: Desingularizing isolated conical singularities: uniform estimates via weighted Sobolev spaces. Commun. Anal. Geom. 21(1), 105–170 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Riesz, F., Sz.-Nagy, B.: Functional analysis. Translated from the second French edition by Leo F. Boron. Reprint of the 1955 original. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York (1990)Google Scholar
  34. 34.
    Schulze, F., Simon, M.: Expanding solitons with non-negative curvature operator coming out of cones. Math. Z. 275(1–2), 625–639 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Simon, M.: Local smoothing results for the Ricci flow in dimensions two and three. Geom. Topol. 17(4), 2263–2287 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tian, G.: Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, Mathematical aspects of string Theory (San Diego, CA, 1986), volume 1 of Advanced Series in Mathematical Physics, pp. 629–646. World Scientific Publishing, Singapore (1987)Google Scholar
  37. 37.
    Tian, G.: K-stability and Kähler-Einstein metrics. Comm. Pure Appl. Math. 68(7), 1085–1156 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Vertman, B.: Ricci flow on singular manifolds, preprint on arXiv:1603.06545 [math.DG] (2016)
  39. 39.
    Vertman, B.: Zeta determinants for regular-singular Laplace-type operators. J. Math. Phys. 50(8), 083515 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wang, Y.: Smooth approximations of the conical Kähler–Ricci flows. Math. Ann. 365(1–2), 835–856 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Yin, H.: Ricci flow on surfaces with conical singularities. J. Geom. Anal. 20(4), 970–995 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University HamburgHamburgGermany
  2. 2.Universität OldenburgOldenburgGermany

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