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 flow by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of nonrotationally symmetric self-shrinkers in ${\mathbf{R}}^3$. Although our initial approximating surfaces are asymptotic to a plane at infinity, the constructed self-similar surfaces are asymptotic to cones at infinity.