Self-shrinking Solutions to Mean Curvature Flow

Abstract

We construct new examples of self-shrinking solutions to mean curvature flow. We first construct an immersed and non-embedded sphere self-shrinker. This result verifies numerical evidence dating back to the 1980's and shows that the rigidity results for constant mean curvature spheres in $\mathbb{R}^3$ and minimal spheres in $S^3$ do not hold for sphere self-shrinkers. Then, in joint work with Stephen Kleene, we construct infinitely many complete, immersed self-shrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus. We also prove rigidity theorems for self-shrinking solutions to geometric flows. In the setting of mean curvature flow, we show that the round sphere is the only embedded sphere self-shrinker with rotational symmetry. In addition, we show that every entire high codimension self-shrinker graph is a plane under a convexity assumption on the angles between the tangent plane to the graph and the base $n$-plane. Finally, in joint work with Peng Lu and Yu Yuan, we show that every complete entire self-shrinking solution on complex Euclidean space to the K\"{a}hler-Ricci flow is generated from a quadratic potential.