Stability of Ricci de Turck flow on singular spaces

Abstract

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.

This is a preview of subscription content, log in to check access.

Fig. 1

Notes

  1. 1.

    If (Mg) is a Ricci-flat space with an isolated conical singularity, then the cross section \((F,g_F)\) of the cone is automatically Einstein with Einstein constant \((n-1)\).

  2. 2.

    The case of \(C=0\) is commonly referred to as linear stability in the literature.

  3. 3.

    Finiteness of the Hölder norm \(\Vert u\Vert _{\alpha }\) in particular implies that u is continuous on the closure \(\overline{M}\) up to the edge singularity, and the supremum may be taken over \((p,p',t) \in \overline{M}^2 \times [0,T]\). Moreover, as explained in [38] we can assume without loss of generality that the tuples \((p,p')\) are always taken from within the same coordinate patch of a given atlas.

  4. 4.

    Differentiation is a priori understood in the distributional sense.

  5. 5.

    We require regularity of \(\omega \) under differentiation by \(x^2\partial _t\) instead of just \(\partial _t\), since in the discussion below, \(\partial _t \omega \mid _{t=0}\) need not be continuous up to \(x=0\).

  6. 6.

    We denote the Schwartz kernel and the fundamental solution by the same letter.

  7. 7.

    The inner product on the symmetric 2-tensors S is defined with respect to g.

  8. 8.

    without convolution in time

  9. 9.

    We do not assume that the conical metric g is Ricci-flat.

  10. 10.

    By Theorem 5.3 its Friedrichs self-adjoint extension is discrete and non-negative.

  11. 11.

    We employ the norms of the Banach space \({{\,\mathrm{\mathscr {H}}\,}}\), unless stated otherwise.

References

  1. 1.

    Bahuaud, E., Vertman, B.: Yamabe flow on manifolds with edges. Math. Nachr. 287(23), 127–159 (2014)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bahuaud, E., Vertman, B..: Long time existence of the edge Yamabe flow, in preparation (2017)

  3. 3.

    Ballmann, W..: Lectures on Kähler manifolds. In: ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS) Zürich (2006)

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Deruelle, A.: Smoothing out positively curved metric cones by Ricci expanders. Geom. Func. Anal. 26, 188–249 (2016)

    MathSciNet  MATH  Google 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)

    Google Scholar 

  10. 10.

    Gallot, S.: Équations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. 12(2), 235–267 (1979)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Giesen, G., Topping, P.M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36(10), 1860–1880 (2011)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler-Einstein metrics with Edge Singularities. Ann. Math. 183(1), 95–176 (2016)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (2006)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Kröncke, K.: Stability of Einstein manifolds. Ph.D. Thesis, Universität Potsdam (2013)

  21. 21.

    Kröncke, K.: Stable and unstable Einstein warped products. Trans. Am. Math. Soc. 369, 6537–6563 (2017)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kröncke, K.: Stability of sin-cones and cosh-cylinders. Ann. Sci. Norm. Super. Pisa, Cl. Sci. 18(3), 1155–1187 (2018)

  23. 23.

    Kühnel, W., Rademacher, H.-B.: Conformal diffeomorphisms preserving the Ricci tensor. Proc. Am. Math. Soc. 123(9), 2841–2849 (1995)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Lichnerowicz, A.: Propagateurs et commutateurs en relativité générale. Publications Mathématiques de l’IHÉS 10(1), 5–56 (1961)

    MATH  Google Scholar 

  25. 25.

    Liu, J., Zhang, X.: Conical Kähler-Ricci flows on Fano manifolds. Adv. Math. 307, 1324–1371 (2017)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Mazzeo, R., Rubinstein, Y., Sesum, N.: Ricci flow on surfaces with conic singularities. Anal. PDE 8(4), 839–882 (2015)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Mazzeo, R., Vertman, B.: Analytic torsion on manifolds with edges. Adv. Math. 231(2), 1000–1040 (2012)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Melrose, R.B.: The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, vol. 4. A K Peters Ltd., Wellesley (1993)

    Google Scholar 

  30. 30.

    Melrose, R.B.: Calculus of conormal distributions on manifolds with corners. Int. Math. Res. Not. 3, 51–61 (1992)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Pacini, T.: Desingularizing isolated conical singularities: uniform estimates via weighted Sobolev spaces. Commun. Anal. Geom. 21(1), 105–170 (2013)

    MathSciNet  MATH  Google 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)

  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)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Simon, M.: Local smoothing results for the Ricci flow in dimensions two and three. Geom. Topol. 17(4), 2263–2287 (2013)

    MathSciNet  MATH  Google 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)

  37. 37.

    Tian, G.: K-stability and Kähler-Einstein metrics. Comm. Pure Appl. Math. 68(7), 1085–1156 (2015)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Wang, Y.: Smooth approximations of the conical Kähler–Ricci flows. Math. Ann. 365(1–2), 835–856 (2016)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Yin, H.: Ricci flow on surfaces with conical singularities. J. Geom. Anal. 20(4), 970–995 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Klaus Kröncke.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Partial support by DFG Priority Programme ”Geometry at Infinity”.

Communicated by M. Struwe.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kröncke, K., Vertman, B. Stability of Ricci de Turck flow on singular spaces. Calc. Var. 58, 74 (2019). https://doi.org/10.1007/s00526-019-1510-7

Download citation

Mathematics Subject Classification

  • Primary 53C44
  • Secondary 53C25
  • 58J35