Fully nonlinear curvature flow of axially symmetric hypersurfaces

  • James A. McCoyEmail author
  • Fatemah Y. Y. Mofarreh
  • Valentina-Mira Wheeler


Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds (J. A. McCoy et al., Annali di Matematica Pura ed Applicata 1–13, 2013). In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (B. H. Andrews, Invent Math 138(1):151–161, 1999; Calc Var Partial Differ Equ 39(3–4):649–657, 2010); these arguments for obtaining a ‘curvature pinching estimate’ may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3.

Mathematics Subject Classification

Primary 35K55 35R35 53C44 Secondary 35K60 


Curvature flow Parabolic partial differential equation Hypersurface Initial-boundary value problem Neumann boundary condition 


  1. 1.
    Altschuler S., Angenent S.B., Giga Y.A.: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5(3), 293–358 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andrews B.H.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2), 151–171 (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews B.H.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151–161 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Andrews B.H.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Andrews B.H.: Moving surfaces by non-concave curvature functions. Calc. Var. Partial Differ. Equ. 39(3–4), 649–657 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Andrews, B.H., Langford, M., McCoy, J.A.: Convexity estimates for fully non-linear surface flows. J. Differ. Geom. (to appear)Google Scholar
  7. 7.
    Andrews B.H., Langford M., McCoy J.A.: Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7(2), 407–433 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Andrews B.H., McCoy J.A.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. Trans. Am. Math. Soc. 364, 3427–3447 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Andrews B.H., McCoy J.A., Zheng Y.: Contracting convex hypersurfaces by curvature. Calc. Var. Partial Differ. Equ. 47(3–4), 611–665 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Athanassenas M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72(1), 52–66 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dzuik G., Kawohl B.: On rotationally symmetric mean curvature flow. J. Differ. Equ. 93(1), 142–150 (1991)CrossRefGoogle Scholar
  12. 12.
    Escher J., Matioc B.-V.: Neck pinching for periodic mean curvature flows. Analysis 30, 253–260 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Huisken G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Huisken G.: Asymptotic behaviour for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Matioc B.-V.: Boundary value problems for rotationally symmetric mean curvature flows. Arch. Math. 89, 365–372 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    McCoy J.A.: Mixed volume preserving curvature flows. Calc. Var. Partial Differ. Equ. 24, 131–154 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    McCoy J.A.: Self-similar solutions of fully nonlinear curvature flows. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10(5), 317–333 (2011)zbMATHMathSciNetGoogle Scholar
  18. 18.
    McCoy, J.A., Mofarreh, F.Y.Y., Williams, G.H.: Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions. Annali di Matematica Pura ed Applicata, 1–13 (2013, to appear in print). doi: 10.1007/s10231-013-0337-7

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • James A. McCoy
    • 1
    Email author
  • Fatemah Y. Y. Mofarreh
    • 1
  • Valentina-Mira Wheeler
    • 1
  1. 1.Institute for Mathematics and its ApplicationsUniversity of WollongongWollongongAustralia

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