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Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

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Abstract

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such support hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.

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Notes

  1. We say a hypersurface M is less than a hypersurface N in \(\varTheta (a_1,a_2)\) if along each vertical line from \(\{z=a_1\}\) to \(\{z=a_2\}\), the intersection point with M is lower than the intersection point with N.

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Acknowledgements

The author is supported by Australian Research Council Discovery Grant DP150100375 at the University of Wollongong. The author is grateful to the Korea Institute for Advanced Study and Hojoo Lee for his hospitality and interesting discussions related to this work.

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  1. Institute for Mathematics and its Applications, University of Wollongong, Northfields Avenue, Wollongong, NSW, 2522, Australia

    Valentina-Mira Wheeler

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  1. Valentina-Mira Wheeler
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Wheeler, VM. Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces. Geom Dedicata 190, 157–183 (2017). https://doi.org/10.1007/s10711-017-0236-y

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