Self-shrinkers with a rotational symmetry

Abstract:

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $\Sigma ^n\subseteq \mathbb {R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\mathbb {R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE.

We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution $\Sigma ^n$ is either a hyperplane $\mathbb {R}^{n}$, the round cylinder $\mathbb {R}\times S^{n-1}$ of radius $\sqrt {2(n-1)}$, the round sphere $S^n$ of radius $\sqrt {2n}$, or is diffeomorphic to an $S^1\times S^{n-1}$ (i.e. a “doughnut” as in the paper by Sigurd B. Angenent, 1992, which when $n=2$ is a torus). In particular, for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.