Moving surfaces by non-concave curvature functions



A convex surface contracting by a strictly monotone, homogeneous degree one function of its principal curvatures remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the concavity of the speed as a function of principal curvatures. We also discuss motion by functions homogeneous of degree greater than 1 in the principal curvatures.

Mathematics Subject Classification (2000)

Primary 53C44 Secondary 35K55 


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  1. 1.
    Andrews B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial. Differ. Equ. 2(2), 151–171 (1994) MR 1385524 (97b:53012)MATHCrossRefGoogle Scholar
  2. 2.
    Andrews, B.: Fully nonlinear parabolic equations in two space variables. available at arXiv: math.DG/0402235Google Scholar
  3. 3.
    Andrews, B.: Gauss curvature flow: the fate of the rolling stones. Invent. Math. 138(1), 151–161 (1999). MR 1714339 (2000i:53097)Google Scholar
  4. 4.
    Andrews, B.: The affine curve-lengthening flow. J. Reine Angew. Math. 506, 43–83 (1999). MR 1665677 (2000e:53081)Google Scholar
  5. 5.
    Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007). MR 2339467 (2008i:53087)Google Scholar
  6. 6.
    Andrews, B., McCoy, J.: Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature. arXiv:0910.0376v1 [math.DG]Google Scholar
  7. 7.
    Chow, B.: Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Differ. Geom. 22(1), 117–138 (1985). MR 826427 (87f:58155)Google Scholar
  8. 8.
    Chow, B.: Deforming convex hypersurfaces by the square root of the scalar curvature. Invent. Math. 87(1), 63–82 (1987). MR 862712 (88a:58204)Google Scholar
  9. 9.
    Glaeser G.: Fonctions composées différentiables. Ann. Math. 77(2), 193–209 (1963) (French). MR 0143058 (26 #624)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984). MR 772132 (86j:53097)Google Scholar
  11. 11.
    Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations. Izv. Akad. Nauk SSSR Ser. Mat. 46(3), 487–523, 670 (1982) (Russian). MR 661144 (84a:35091)Google Scholar
  12. 12.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co. Inc., River Edge, NJ, (1996). MR 1465184 (98k:35003)Google Scholar
  13. 13.
    Schulze, F.: Convexity estimates for flows by powers of the mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5(2), 261–277 (2006). MR 2244700 (2007b:53138)Google Scholar
  14. 14.
    Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975) MR 0370643 (51 #6870)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tso, K.: Deforming a hypersurface by its Gauss–Kronecker curvature. Commun. Pure Appl. Math. 38(6), 867–882 (1985). MR 812353 (87e:53009)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.MSI, ANUCanberraAustralia

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