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Evolution of convex hypersurfaces by powers of the mean curvature

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Abstract

We study the evolution of a closed, convex hypersurface in ℝn+1 in direction of its normal vector, where the speed equals a positive power k of the mean curvature. We show that the flow exists on a maximal, finite time interval, and that, approaching the final time, the surfaces contract to a point.

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References

  1. Andrews, B.: Contraction of convex hypersurfaces in Euclidian space. Calc. Var. 2, 151–171 (1994)

    Article  Google Scholar 

  2. Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pacific J. Math 195(1), 1–36 (2000)

    Google Scholar 

  3. Chow, B.: Deforming convex hypersurfaces by the nth root of the Gaussian curvature. J. Diff. Geom. 22, 117–138 (1985)

    Google Scholar 

  4. Chow, B.: Deforming hypersurfaces by the square root of the scalar curvature. Invent. Math. 87, 63–82 (1987)

    Article  Google Scholar 

  5. Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Diff. Geom. 32(1), 299–314 (1990)

    Google Scholar 

  6. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom. 20, 237–266 (1984)

    Google Scholar 

  7. Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. Calc. of Var. and Geom. Evo. Probl., CIME Lectures of Cetraro, Springer, 1996

  8. Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)

    Google Scholar 

  9. Krylov, N.V.: Nonlinear elliptic and parabolic equations of second order. D. Reidel, 1978

  10. Lieberman, G.M.: Second order parabolic differential equations. World Scientific, 1996

  11. Tso, K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Comm. Pure Appl. Math. 38, 867–882 (1985)

    Google Scholar 

  12. Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(3), 355–372 (1990)

    Google Scholar 

  13. Urbas, J.: An expansion of convex hypersurfaces. J. Diff. Geom. 33(1), 91–125 (1991)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, 8092, Zürich, Switzerland

    Felix Schulze

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  1. Felix Schulze
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Correspondence to Felix Schulze.

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The author was partially supported by a Schweizerische Nationalfonds grant No. 21-66743.01.

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Schulze, F. Evolution of convex hypersurfaces by powers of the mean curvature. Math. Z. 251, 721–733 (2005). https://doi.org/10.1007/s00209-004-0721-5

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  • DOI: https://doi.org/10.1007/s00209-004-0721-5

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