The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1555–1570 | Cite as

New Pinching Estimates for Inverse Curvature Flows in Space Forms

  • Yong WeiEmail author


We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature \(K_N\) with speed given by \(F^{-\alpha }\), where \(\alpha \in (0,1]\) for \(K_N=0,-1\) and \(\alpha =1\) for \(K_N=1\), F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual \(F_*\) approaching zero on the boundary of the positive cone \(\Gamma _+\). We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.


Pinching estimate Inverse curvature flow Space form Inverse concave 

Mathematics Subject Classification

53C44 53C21 



The author would like to thank the referees for carefully reading of this manuscript and providing many helpful suggestions. The author was supported by Ben Andrews throughout his Australian Laureate Fellowship FL150100126 of the Australian Research Council.


  1. 1.
    Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2), 151–171 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, B.: Contraction of convex hypersurfaces in Riemannian spaces. J. Differ. Geom. 39(2), 407–431 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pacific. J. Math. 195(1), 1–34 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrews, B.: Fully nonlinear parabolic equations in two space variables. arXiv: math.DG/0402235, (2004)
  5. 5.
    Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Andrews, B., Langford, M., McCoy, J.: Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7(2), 407–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, B., McCoy, J., Zheng, Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Partial Differ. Equ. 47(3–4), 611–665 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Andrews, B., Wei, Y.: Quermassintegral preserving curvature flow in Hyperbolic space. Geom. Funct. Anal (to appear). arXiv:1708.09583
  9. 9.
    Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gerhardt, C.: Curvature problems, Series in Geometry and Topology, vol. 39. International Press, Somerville, MA (2006)zbMATHGoogle Scholar
  12. 12.
    Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89(3), 487–527 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gerhardt, C.: Non-scale-invariant inverse curvature flows in Euclidean space. Calc. Var. Partial Differ. Equ. 49(1–2), 471–489 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gerhardt, C.: Curvature flows in the sphere. J. Differ. Geom. 100(2), 301–347 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487–523, 670 (1982). RussianMathSciNetzbMATHGoogle Scholar
  17. 17.
    Kröner, H., Scheuer, J.: Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature. arXiv:1703.07087
  18. 18.
    Li, H., Wang, X., Wei, Y.: Surfaces expanding by non-concave curvature functions. arXiv:1609.00570
  19. 19.
    McCoy, J.A.: More mixed volume preserving curvature flows. J. Geom. Anal. 27(4), 3140–3165 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Makowski, M., Scheuer, J.: Rigidity results, inverse curvature flows and alexandrov-fenchel type inequalities in the sphere. Asian J. Math. 20(5), 869–892 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Scheuer, J.: Gradient estimates for inverse curvature flows in hyperbolic space. Geom. Flows 1(1), 11–16 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Scheuer, J.: Non-scale-invariant inverse curvature flows in hyperbolic space. Calc. Var. Partial Differ. Equ. 53(1–2), 91–123 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Scheuer, J.: Pinching and asymptotical roundness for inverse curvature flows in Euclidean space. J. Geom. Anal. 26(3), 2265–2281 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Urbas, J.I.E.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(1), 355–372 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Urbas, J.I.E.: An expansion of convex hypersurfaces. J. Differ. Geom. 33(1), 91–125 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

Personalised recommendations