Paper · Jun 17, 2024

Embedded cylindrical and doughnut-shaped $λ$ -hypersurfaces

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Abstract

In the paper, we construct, for $λ > 0$ , complete embedded and non-convex $λ$ -hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $λ$ -hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle affirmatively. Furthermore, for a fixed $λ < 0$ which may have small $∣ λ ∣$ , we can construct two compact embedded $λ$ -hypersurfaces which are diffeomorphic to $S^{1} \times S^{n - 1}$ , but they are not isometric to each other.

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Mean curvature flow Self-shrinkers & translators

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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)

In the guidebook

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.

Embedded cylindrical and doughnut-shaped $λ$ -hypersurfaces

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Embedded cylindrical and doughnut-shaped λ-hypersurfaces

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Embedded cylindrical and doughnut-shaped $λ$ -hypersurfaces