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

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.