Paper · Aug 31, 2025

Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

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Abstract

In this paper, we investigate the convexity of mean convex asymptotically conical self-expanders to the mean curvature flow in $R^{n + 1}$ . Specifically, for $n \geq 3$ , we show that any $n$ -dimensional complete mean convex self-expander asymptotic to mean convex and weakly convex cones must be strictly convex.

Comments: 19 pages

Mean curvature flow Self-shrinkers & translators Boundaries & non-compact

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

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