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Inverse Mean Curvature Flow for Star-Shaped Hypersurfaces Evolving in a Cone

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Abstract

For a given convex cone we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicular. The evolution of those hypersurfaces inside the cone yields a nonlinear parabolic Neumann problem. We show that one can use the convexity of the cone to prove long time existence of this flow. Finally, we show that the hypersurfaces converge smoothly to a piece of the round sphere.

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Notes

  1. That means the second fundamental form of ∂M n is positive definite with respect to the outward unit normal n∈T x M n∩N x ∂M n.

  2. In the whole article C 2k+α,k+α/2 denote the parabolic Hölder spaces as they are defined in [8], but we use the letter C instead of H. Furthermore, we use the Einstein summation convention to sum over repeated indices.

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Acknowledgements

The author wants to thank Professor Huisken for acquainting him with inverse mean curvature flow, Professor Schnürer for suggesting the problem of the evolution in a cone, and both for helpful discussions.

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  1. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476, Potsdam-Golm, Germany

    Thomas Marquardt

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  1. Thomas Marquardt
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Correspondence to Thomas Marquardt.

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Marquardt, T. Inverse Mean Curvature Flow for Star-Shaped Hypersurfaces Evolving in a Cone. J Geom Anal 23, 1303–1313 (2013). https://doi.org/10.1007/s12220-011-9288-7

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  • DOI: https://doi.org/10.1007/s12220-011-9288-7

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