Geometric and Functional Analysis

, Volume 28, Issue 5, pp 1183–1208 | Cite as

Quermassintegral preserving curvature flow in Hyperbolic space

  • Ben AndrewsEmail author
  • Yong Wei


We consider the quermassintegral preserving flow of closed h-convex hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function f of the principal curvatures which is inverse concave and has dual f* approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is h-convex, then the solution of the flow becomes strictly h-convex for t > 0, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

Keywords and phrases

Quermassintegral preserving flow Hyperbolic space Alexandrov reflection 

Mathematics Subject Classification

53C44 53C21 


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  1. And94.
    Andrews B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ., (2)2, 151–171 (1994)MathSciNetCrossRefGoogle Scholar
  2. And00.
    Andrews B.: Motion of hypersurfaces by Gauss curvature. Pacific. J. Math., (1) 195, 1–34 (2000)MathSciNetCrossRefGoogle Scholar
  3. And01.
    Andrews B.: Volume-preserving anisotropic mean curvature flow. Indiana Univ. Math. J., (2)50, 783–827 (2001)MathSciNetCrossRefGoogle Scholar
  4. And07.
    Andrews B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)MathSciNetzbMATHGoogle Scholar
  5. AC17.
    Andrews B., Chen X.: Curvature flow in hyperbolic spaces. J. Reine. Angew. Math. 729, 29–49 (2017)MathSciNetzbMATHGoogle Scholar
  6. ALM14.
    Andrews B., Langford M., McCoy J.: Convexity estimates for hyper surfaces moving by convex curvature functions. Analysis and PDE, (2)7, 407–433 (2014)MathSciNetCrossRefGoogle Scholar
  7. AMZ13.
    Andrews B., McCoy J., Zheng Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Partial Differ. Equ. (3–4)47, 611–665 (2013)MathSciNetCrossRefGoogle Scholar
  8. AW17.
    B. Andrews and Y. Wei. Volume preserving flow by powers of k-th mean curvature. arXiv:1708.03982.
  9. BM99.
    Borisenko A., Miquel V.: Total curvature of convex hypersurfaces in the hyperbolic space. Illinois J. Math., (1)43, 61–78 (1999)MathSciNetzbMATHGoogle Scholar
  10. BM02.
    Borisenko A., Miquel V.: Comparison theorems on convex hypersurfaces in Hadamard manifolds. Ann. Global Anal. Geom. (2)21, 191–202 (2002)MathSciNetCrossRefGoogle Scholar
  11. BS.
    M.C. Bertini and C. Sinestrari. Volume preserving non homogeneous mean curvature flow of convex hypersurfaces, to appear on Annali di Matematica Pura ed Applicata. Scholar
  12. BP17.
    Bertini M.C., Pipoli G.: Volume preserving non homogeneous mean curvature flow in hyperbolic space. Diff. Geom. Appl. 54, 448–463 (2017)MathSciNetCrossRefGoogle Scholar
  13. CM07.
    Cabezas-Rivas E., Miquel V.: Volume preserving mean curvature flow in the hyperbolic space. Indiana Univ. Math. J. 56, 2061–2086 (2007)MathSciNetCrossRefGoogle Scholar
  14. CS10.
    Cabezas-Rivas E., Sinestrari C.: Volume-preserving flow by powers of the mth mean curvature. Calc. Var. Partial Differ. Equ. 38, 441–469 (2010)CrossRefGoogle Scholar
  15. Ger06.
    Gerhardt C.: Curvature problems, Series in Geometry and Topology, vol.39 International Press. International Press, Somerville M.A (2006)Google Scholar
  16. GL15.
    Guan P., Li J.: A mean curvature type flow in space forms. Int Math Res Notices, (13)2015, 4716–4740 (2015)MathSciNetCrossRefGoogle Scholar
  17. GLW17.
    Guo S., Li G., Wu C.: Volume preserving flow by power of the mth mean curvature in the hyperbolic space. Comm. Anal. Geom., (2)25, 321–372 (2017)MathSciNetCrossRefGoogle Scholar
  18. GWW14.
    Ge Y., Wang G., Wu J.: Hyperbolic Alexandrov-Fenchel quermass integral inequalities II. J. Differential Geom. (2)98, 237–260 (2014)MathSciNetCrossRefGoogle Scholar
  19. Ham82.
    Hamilton R.S.: Three-manifolds with positive Ricci curvature. J.Differential. Geom., (2)17, 255–306 (1982)MathSciNetCrossRefGoogle Scholar
  20. Hui87.
    Huisken G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)MathSciNetzbMATHGoogle Scholar
  21. Kry82.
    Krylov N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. (3)46, 487–523 (1982) 670 (Russian).MathSciNetzbMATHGoogle Scholar
  22. Lie96.
    Lieberman Gary M.: Second order parabolic differential equations. World scientific, Singapore (1996)CrossRefGoogle Scholar
  23. LWX14.
    Li H., Wei Y., Xiong C.: A geometric inequality on hypersurface in hyperbolic space. Adv. Math. 253, 152–162 (2014)MathSciNetCrossRefGoogle Scholar
  24. McC03.
    McCoy J.A.: The surface area preserving mean curvature flow. Asian J. Math. 7, 7–30 (2003)MathSciNetCrossRefGoogle Scholar
  25. McC04.
    McCoy J.A.: The mixed volume preserving mean curvature flow. Math. Z., (1)246, 155–166 (2004)MathSciNetCrossRefGoogle Scholar
  26. McC05.
    McCoy J.A.: Mixed volume preserving curvature flows. Calc. Var. Partial Differ. Equ. 24, 131–154 (2005)MathSciNetCrossRefGoogle Scholar
  27. McC17.
    McCoy J.A.: More mixed volume preserving curvature flows. J. Geom. Anal., (4)27, 3140–3165 (2017)MathSciNetCrossRefGoogle Scholar
  28. Mak12.
    M. Makowski. Mixed volume preserving curvature flows in hyperbolic space. arXiv:1208.1898.
  29. Sin15.
    Sinestrari C.: Convex hypersurfaces evolving by volume preserving curvature flows. Calc. Var. Partial Differ. Equ. 54, 1985–1993 (2015)MathSciNetCrossRefGoogle Scholar
  30. San04.
    Santalo L.A.: Integral geometry and geometric probability Second edition.With a foreword by Mark Kac. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)Google Scholar
  31. Sol06.
    Solanes G.: Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces. Trans. Amer. Math. Soc., (3)358, 1105–1115 (2006)MathSciNetCrossRefGoogle Scholar
  32. Tso85.
    Tso K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38, 867–882 (1985)MathSciNetCrossRefGoogle Scholar
  33. WX14.
    Wang G., Xia C.: Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space. Adv. Math. 259, 532–556 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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