Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1206–1232 | Cite as

Ancient Solutions of Geometric Flows with Curvature Pinching

  • Susanna Risa
  • Carlo SinestrariEmail author
Article

Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.

Keywords

Ancient solutions Mean curvature flow Gauss curvature flow Geometric flows 

Mathematics Subject Classification

53C44 35K55 

Notes

Acknowledgements

Carlo Sinestrari was partially supported by the research group GNAMPA of INdAM (Istituto Nazionale di Alta Matematica).

References

  1. 1.
    Alessandroni, R., Sinestrari, C.: Evolution of hypersurfaces by powers of the scalar curvature. Ann.Sc. Norm. Super. Pisa Cl. Sci. Ser. V 9(3), 541–571 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Part. Differ. Equat. 2(2), 151–171 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, B.: Harnack inequalities for evolving hypersurfaces. Math. Z. 217(2), 179–197 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrews, B.: Contraction of convex hypersurfaces by their affine normal. J. Differ. Geom. 43(2), 207–230 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andrews, B.: Noncollapsing in mean-convex mean curvature flow. Geom. Topol. 16(3), 1413–1418 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Andrews, B., Baker, C.: Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85(3), 357–395 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, B., Guan, P., Ni, L.: Flow by powers of the Gauss curvature. Adv. Math. 299, 174–201 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Andrews, B., McCoy, J.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Trans. Am. Math Soc. 364(7), 3427–3447 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Andrews, B., McCoy, J., Zheng, Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Part. Differ. Equat. 47(3–4), 611–665 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Angenent, S.: Shrinking doughnuts. Progr. Nonlinear Differ. Equat. Appl. 7, 21–38 (1992)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Angenent, S.: Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Netw. Heterog. Media 8(1), 1–8 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Angenent, S., Daskalopoulos, P., Sesum, N.: Unique asymptotics of ancient convex mean curvature flow solutions. arXiv:1503.01178 (2015)
  13. 13.
    Bakas, I., Sourdis, C.: Dirichlet sigma models and mean curvature flow. J. High Energy Phys. 06, 057 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Baker, C.: The mean curvature flow of submanifolds of high codimension (PhD thesis). arXiv:1104.4409 (2011)
  15. 15.
    Bourni, T., Langford, M., Tinaglia, G.: A collapsing ancient solution of mean curvature flow in \({\mathbb{R}}^3\). arXiv:1705.06981 (2011)
  16. 16.
    Brendle, S., Choi, K., Daskalopoulos, P., et al.: Asymptotic behavior of flows by powers of the Gaussian curvature. Acta Math. 219(1), 1–16 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bryan, P., Ivaki, M., Scheuer, J.: On the classification of ancient solutions to curvature flows on the sphere. arXiv:1604.01694 (2016)
  19. 19.
    Bryan, P., Louie, J.: Classification of convex ancient solutions to curve shortening flow on the sphere. J. Geom. Anal. 26(2), 858–872 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cabezas-Rivas, E., Sinestrari, C.: Volume-preserving flow by powers of the mth mean curvature. Calc. Var. Part. Differ. Equat. 38(3), 441–469 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60(6), 568–578 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Choi, K., Daskalopoulos, P.: Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487 (2016)
  23. 23.
    Chow, B.: Deforming convex hypersurfaces by the \( n \) th root of the Gaussian curvature. J. Differ Geom. 22(1), 117–138 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chow, B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87(1), 63–82 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Chow, B.: On Harnack’s inequality and entropy for the Gaussian curvature flow. Commun. Pure Appl. Math. 44(4), 469–483 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of compact ancient solutions to the curve shortening flow. J. Differ. Geom. 84, 455–464 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  28. 28.
    DiBenedetto, E., Friedman, A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357(4), 1–22 (1985)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Firey, W.J.: Shapes of worn stones. Mathematika 21(1), 1–11 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gerhardt, C.: Curvature Problems, Series in Geometry and Topology, vol. 39. International Press, Somerville (2006)zbMATHGoogle Scholar
  31. 31.
    Guan, P., Ni, L.: Entropy and a convergence theorem for Gauss curvature flow in high dimension. J. Eur. Math. Soc. 19(12), 3735–3761 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hamilton, R.: Formation of singularities in the Ricci flow. Surv Differ. Geom. 2, 7–136 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Haslhofer, R., Hershkovits, O.: Ancient solutions of the mean curvature flow. Commun. Anal. Geom. 24(3), 593–604 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. 70(3), 511–546 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Huisken, G., Sinestrari, C.: Convex ancient solutions of the mean curvature flow. J. Differ. Geom. 101(2), 267–287 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order, vol. 7. Springer, Norwell (1987)CrossRefzbMATHGoogle Scholar
  39. 39.
    Langford, M.: A general pinching principle for mean curvature flow and applications. Calc. Var. Part. Differ. Equat. 56(4), 107 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Langford, M., Lynch, S.: Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions. arXiv:1704.03802 (2017)
  41. 41.
    Lukyanov, S., Vitchev, E., Zamolodchikov, A.: Integrable model of boundary interaction: the paperclip. J. Nucl. Phys. B 683, 423–454 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lynch, S., Nguyen, H.T.: Pinched ancient solutions to the high codimension mean curvature flow. arXiv:1709.09697 (2017)
  43. 43.
    Pipoli, G., Sinestrari, C.: Mean curvature flow of pinched submanifolds of CPn. Commun. Anal. Geom. 25(4), 799–846 (2017)CrossRefzbMATHGoogle Scholar
  44. 44.
    Schulze, F.: Convexity estimates for flows by powers of the mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 261–277 (2006)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Sinestrari, C.: Convex hypersurfaces evolving by volume preserving curvature flows. Calc. Var. Part. Differ Equat. 54(2), 1985–1993 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tso, K.: Deforming a hypersurface by its Gauss–Kronecker curvature. Commun. Pure Appl. Math. 38(6), 867–882 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16(1), 123–138 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Dipartimento di Ingegneria Civile e Ingegneria InformaticaUniversità di Roma “Tor Vergata”RomaItaly

Personalised recommendations