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Minimal cones and self-expanding solutions for mean curvature flows

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In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. Ilmanen (Lectures on mean curvature flow and related equations (Trieste Notes), 1995) proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is \(C^{3,\alpha }\)-regular and mean convex (but not area-minimizing), we can prove that the corresponding self-expanding hypersurfaces are smooth, embedded, and have positive mean curvature everywhere (see Theorem 1.1). As a result, for regular minimal but not area-minimizing cones we can give an affirmative answer to a problem arisen by Lawson (Brothers, in Proc Sympos Pure Math 44:441–464, 1986).

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Acknowledgements

The author would like to express his gratitude to Prof. Y. L. Xin for inspiring discussion and valuable advice, and to Prof. J. Jost for his constant encouragement and support. He would also like to thank Prof. F. Morgan for his interests and suggestions. The author would like to express his sincere gratitude to the referees for valuable comments that will help to improve the quality of the manuscript. The author is supported partially by Natural Science Foundation of Shanghai (Grant No. 15ZR1402200) and Shanghai Municipal Education Commission.

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  1. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Qi Ding

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  1. Qi Ding
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Correspondence to Qi Ding.

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Communicated by F.C. Marques.

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Ding, Q. Minimal cones and self-expanding solutions for mean curvature flows. Math. Ann. 376, 359–405 (2020). https://doi.org/10.1007/s00208-019-01941-1

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