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

TL;DR

This paper constructs new complete embedded self-similar surfaces under mean curvature flow by gluing a sphere and a plane, resulting in non-rotationally symmetric self-shrinkers with finite genus.

Contribution

It introduces the first known examples of non-rotationally symmetric self-shrinkers in b2^3 by a novel gluing and perturbation method.

Findings

Constructed new self-similar surfaces asymptotic to cones at infinity.

Produced a one-parameter family of self-shrinkers.

Surfaces have finite genus and are embedded.

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.