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Minimal cones and self-expanding solutions for mean curvature flows

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-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 [4].