Minimal cones and self-expanding solutions for mean curvature flows

Abstract

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].

The space of asymptotically conical self-expanders of mean curvature flow

“…They are also expected to model the long time behavior of the flow. The interested reader may refer to [2], [7], [8], [9], [11], [15], [16], [20], [21], and references therein. Finally, self-expanders arise variationally as stationary points, with respect to compactly supported variations, of the functional…”

Self-Expanders of the Mean Curvature Flow

“…Fong and McGrath [8] proved a Liouville-type theorem for complete, mean-convex self-expanders whose ends have decaying principal curvatures. Ding [6] studied self-expanding solutions and their relationship to minimal cones. The space of asymptotically conical self-expanders was studied in several papers by Bernstein and Wang, as for example in [2] and [3].…”

Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

“…It is known that the singular minimal cones are the singular self-expanders. Recently, Ding [10] obtained some results on minimal cones and self-expanders. There are other works in self-expanders (see, e.g.…”

“…L-stability (see Section 2) means that the second variation of its weighted volume is nonnegative for any compactly supported normal variation (see, e.g., [3], [10]). Recall that there is no weighted-stable self-shrinkers with polynomial volume growth ( [9]).…”