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Ancient Solutions of Geometric Flows with Curvature Pinching

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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.

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References

  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)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Part. Differ. Equat. 2(2), 151–171 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews, B.: Harnack inequalities for evolving hypersurfaces. Math. Z. 217(2), 179–197 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews, B.: Contraction of convex hypersurfaces by their affine normal. J. Differ. Geom. 43(2), 207–230 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrews, B.: Noncollapsing in mean-convex mean curvature flow. Geom. Topol. 16(3), 1413–1418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrews, B., Baker, C.: Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85(3), 357–395 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrews, B., Guan, P., Ni, L.: Flow by powers of the Gauss curvature. Adv. Math. 299, 174–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Andrews, B., McCoy, J., Zheng, Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Part. Differ. Equat. 47(3–4), 611–665 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Angenent, S.: Shrinking doughnuts. Progr. Nonlinear Differ. Equat. Appl. 7, 21–38 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Angenent, S.: Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Netw. Heterog. Media 8(1), 1–8 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Angenent, S., Daskalopoulos, P., Sesum, N.: Unique asymptotics of ancient convex mean curvature flow solutions. arXiv:1503.01178 (2015)

  13. Bakas, I., Sourdis, C.: Dirichlet sigma models and mean curvature flow. J. High Energy Phys. 06, 057 (2007)

    Article  MathSciNet  Google Scholar 

  14. Baker, C.: The mean curvature flow of submanifolds of high codimension (PhD thesis). arXiv:1104.4409 (2011)

  15. Bourni, T., Langford, M., Tinaglia, G.: A collapsing ancient solution of mean curvature flow in \({\mathbb{R}}^3\). arXiv:1705.06981 (2011)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bryan, P., Ivaki, M., Scheuer, J.: On the classification of ancient solutions to curvature flows on the sphere. arXiv:1604.01694 (2016)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  21. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60(6), 568–578 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Choi, K., Daskalopoulos, P.: Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487 (2016)

  23. Chow, B.: Deforming convex hypersurfaces by the \( n \) th root of the Gaussian curvature. J. Differ Geom. 22(1), 117–138 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chow, B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87(1), 63–82 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chow, B.: On Harnack’s inequality and entropy for the Gaussian curvature flow. Commun. Pure Appl. Math. 44(4), 469–483 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Daskalopoulos, P., Hamilton, R., Sesum, N.: Classification of compact ancient solutions to the curve shortening flow. J. Differ. Geom. 84, 455–464 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  28. DiBenedetto, E., Friedman, A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357(4), 1–22 (1985)

    MathSciNet  MATH  Google Scholar 

  29. Firey, W.J.: Shapes of worn stones. Mathematika 21(1), 1–11 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gerhardt, C.: Curvature Problems, Series in Geometry and Topology, vol. 39. International Press, Somerville (2006)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hamilton, R.: Formation of singularities in the Ricci flow. Surv Differ. Geom. 2, 7–136 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Haslhofer, R., Hershkovits, O.: Ancient solutions of the mean curvature flow. Commun. Anal. Geom. 24(3), 593–604 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. Commun. Pure Appl. Math. 70(3), 511–546 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Huisken, G., Sinestrari, C.: Convex ancient solutions of the mean curvature flow. J. Differ. Geom. 101(2), 267–287 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order, vol. 7. Springer, Norwell (1987)

    Book  MATH  Google Scholar 

  39. Langford, M.: A general pinching principle for mean curvature flow and applications. Calc. Var. Part. Differ. Equat. 56(4), 107 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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. Lukyanov, S., Vitchev, E., Zamolodchikov, A.: Integrable model of boundary interaction: the paperclip. J. Nucl. Phys. B 683, 423–454 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lynch, S., Nguyen, H.T.: Pinched ancient solutions to the high codimension mean curvature flow. arXiv:1709.09697 (2017)

  43. Pipoli, G., Sinestrari, C.: Mean curvature flow of pinched submanifolds of CPn. Commun. Anal. Geom. 25(4), 799–846 (2017)

    Article  MATH  Google Scholar 

  44. Schulze, F.: Convexity estimates for flows by powers of the mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 261–277 (2006)

    MathSciNet  MATH  Google Scholar 

  45. Sinestrari, C.: Convex hypersurfaces evolving by volume preserving curvature flows. Calc. Var. Part. Differ Equat. 54(2), 1985–1993 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tso, K.: Deforming a hypersurface by its Gauss–Kronecker curvature. Commun. Pure Appl. Math. 38(6), 867–882 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16(1), 123–138 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italy

    Susanna Risa

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma “Tor Vergata”, Via Politecnico 1, 00133, Roma, Italy

    Carlo Sinestrari

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  1. Susanna Risa
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  2. Carlo Sinestrari
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Correspondence to Carlo Sinestrari.

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Risa, S., Sinestrari, C. Ancient Solutions of Geometric Flows with Curvature Pinching. J Geom Anal 29, 1206–1232 (2019). https://doi.org/10.1007/s12220-018-0036-0

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  • DOI: https://doi.org/10.1007/s12220-018-0036-0

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