Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part III

Abstract

We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in ${\mathbb{R}}^3$ that are not rotationally symmetric. The strategy for the construction is to start with a family of initial surfaces by desingularizing the intersection of a sphere and a plane, then solve a perturbation problem to obtain a one parameter family of self-similar surfaces. Although we start with surfaces asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.

26 pages, typos corrected. To appear in Duke Mathematical Journal

References in corpus (1)

• Complete Embedded Self-Translating Surfaces under Mean Curvature Flow

Cited by in corpus (9)

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• Bi-Halfspace and Convex Hull Theorems for Translating Solitons
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• Noncompact self-shrinkers for mean curvature flow with arbitrary genus
• Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori
• Genus one singularities in mean curvature flow
• Entropy Bounds, Compactness and Finiteness Theorems for Embedded Self-shrinkers with Rotational Symmetry
• Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
• Ancient mean curvature flows from minimal hypersurfaces