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Contracting Convex Hypersurfaces in Space Form by Non-homogeneous Curvature Function

  • Guanghan Li
  • Yusha LvEmail author
Article
  • 17 Downloads

Abstract

This paper concerns the evolution of closed convex hypersurfaces in space form \({\mathbb {N}}^{n+1}_\kappa \) of constant sectional curvature \(\kappa =0, \pm 1\) by curvature flow, for which the normal speed is given by a general non-homogeneous function \(\varPhi \) of curvature function F which is monotone, symmetric, homogeneous of degree one. Under the assumption that F is inverse concave and its dual function \(F_*\) approaches zero on the boundary of positive cone, we prove that the flow converges to a point in finite time. For special flow with \(\varPhi =F^\beta ~(\beta >1)\), without the assumption of inverse concavity on F, we deduce that if the initial hypersurface is pinched enough, then the flow converges smoothly and exponentially to the unit sphere of \({\mathbb {R}}^{n+1}\) after some suitable rescaling.

Keywords

Non-homogeneous curvature flow Convex hypersurfaces Space form 

Mathematics Subject Classification

53C44 35K55 

Notes

Acknowledgements

The authors would like to thank the referees for careful reading of the manuscript and their valuable comments.

References

  1. 1.
    Aldaz, J.: A refinement of the inequality between arithmetic and geometric means. J. Math. Inequal. 2, 473–477 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldaz, J.: Self-improvement of the inequality between arithmetic and geometric means. J. Math. Inequal. 3, 213–216 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alessandroni, R.: Evolution of hypersurfaces by curvature functions, Ph.D. Thesis, Università degli studi di Roma “Tor Vergata” (2008). http://dspace.uniroma2.it/dspace/handle/2108/661
  4. 4.
    Alessandroni, R., Sinestrari, C.: Evolution of hypersurfaces by powers of the scalar curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. 9, 541–571 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Alessandroni, R., Sinestrari, C.: Convexity estimates for a nonhomogeneous mean curvature flow. Math. Z. 266, 65–82 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2, 151–171 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Differ. Geom. 39, 407–431 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Andrews, B., Chen, X., Wei, Y.: Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space. arXiv:1805.11776
  10. 10.
    Andrews, B., Langford, M., McCoy, J.: Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7, 407–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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, 3427–3447 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Andrews, B., McCoy, J., Zheng, Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Partial Differ. Equ. 47, 611–665 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Andrews, B., Wei, Y.: Volume preserving flow by powers of \(k\)-th mean curvature. arXiv:1708.03982
  14. 14.
    Andrews, B., Wei, Y.: Quermassintegral preserving curvature flow in hyperboic space. Geom. Funct. Anal. 28, 1183–1208 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bertini, M., Pipoli, G.: Volume preserving non homogeneous mean curvature flow in hyperbolic space. Differ. Geom. Appl. 54, 448–463 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bertini, M., Sinestrari, C.: Volume-preserving nonhomogeneous mean curvature flow of convex hypersurfaces. Ann. Mat. Pura Appl. 197, 1295–1309 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bertini, M., Sinestrari, C.: Volume preserving flow by powers of symmetric polynomials in the principal curvatures. Math. Z. 289, 1219–1236 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Borisenko, A., Miquel, V.: Total curvatures of convex hypersurfaces in hyperbolic space. Ill. J. Math. 43, 61–78 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Brendle, S., Choi, K., Daskalopoulos, P.: Asymptotic behavior of flows by powers of the Gaussian curvature. Acta. Math. 219, 1–16 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cabezas-Rivas, E., Sinestrari, C.: Volume-preserving flow by powers of the mth mean curvature. Calc. Var. Partial Differ. Equ. 38, 441–469 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Caffarelli, L.: Interior a priori estimates for solutions of fully non-linear equations. Ann. Math. 130, 135–150 (1989)CrossRefGoogle Scholar
  22. 22.
    Chow, B.: Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Differ. Geom. 22, 117–138 (1985)CrossRefzbMATHGoogle Scholar
  23. 23.
    Chow, B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87, 63–82 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chow, B., Tsai, D.: Expansion of convex hypersurface by non-homogeneous functions of curvature. Asian J. Math. 1, 769–784 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Cordes, H.: Über die erste Randwertaufgabe bei quasilinearen Differential-gleichungen zweiter Ordnung in mehr als zwei Variablen. Math. Ann. 131, 278–312 (1956). (German)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gerhardt, C.: Curvature Problems. Series in Geometry and Topology. International Press, Somerville, MA (2006)zbMATHGoogle Scholar
  27. 27.
    Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geom. 100, 301–347 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Guo, S.: Contracting convex hypersurfaces by functions of the mean curvature. arXiv:1610.08209
  29. 29.
    Guo, S., Li, G., Wu, C.: Contraction of horosphere-convex hypersurfaces by powers of the mean curvature in the hyperbolic space. J. Korean Math. Soc. 50, 1311–1332 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Guo, S., Li, G., Wu, C.: Deforming pinched hypersurfaces of the hyperbolic space by powers of the mean curvature into spheres. J. Korean Math. Soc. 53, 737–767 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Guo, S., Li, G., Wu, C.: Volume-preserving flow by powere of the \(m\)-th mean curvature in the hyperbolic space. Commun. Anal. Geom. 25, 321–372 (2017)CrossRefzbMATHGoogle Scholar
  32. 32.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84, 463–480 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Huisken, G., Polden, A.: Geometric Evolution Equations for Hypersurfaces, Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, pp. 45–84. Springer, Berlin (1999)zbMATHGoogle Scholar
  36. 36.
    Krylov, N., Safonov, M.: A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk. 40, 161-175 (1980) English Transl. Math. USSR Izv. 16, 151–164 (1981)Google Scholar
  37. 37.
    Li, G., Lv, Y.: Flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space. J. Math. Anal. Appl. 460, 808–837 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge, NJ (1996)CrossRefzbMATHGoogle Scholar
  39. 39.
    Nirenberg, L.: On a Generalization of Quasi-Conformal Mappings and Its Application to Elliptic Partial Differential Equations, Contributions to the Theory of Partial Differential Equations. Annals of Mathematics Studies, pp. 95–100. Princeton University Press, Princeton, NJ (1954)Google Scholar
  40. 40.
    Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. No. 151. Cambridge University Press, Cambridge (2014)Google Scholar
  41. 41.
    Schulze, F.: Evolution of convex hypersurfaces by powers of the mean curvature. Math. Z. 251, 721–733 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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
  43. 43.
    Smoczyk, K.: Harnack inequalities for curvature flows depending on mean curvature. N. Y. J. Math. 3, 103–118 (1997)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Smoczyk, K.: Starshaped hypersurfaces and the mean curvature flow. Manuscr. Math. 95, 225–236 (1998)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Tso, K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38, 867–882 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wei, Y.: New pinching estimates for inverse curvature flow in space form. J. Geom. Anal. (2018).  https://doi.org/10.1007/s12220-018-0051-1
  47. 47.
    Zhu, X.: Lectures on Mean Curvature Flows. AMS/IP Studies in Advanced Mathematics, vol. 32. American Mathematical Society, Providence, RI; International Press, Somerville, MA (2002)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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