Self-shrinkers with a rotational symmetry

TL;DR

This paper introduces a new family of rotationally symmetric self-shrinking hypersurfaces in Euclidean space, classifies all such embedded hypersurfaces, and employs global analysis of a cubic-derivative ODE for proofs.

Contribution

It presents a novel family of rotational self-shrinkers with specific geometric properties and provides a complete classification of embedded rotational self-shrinking hypersurfaces.

Findings

New family of self-shrinkers interpolating between plane and half-cylinder.

Complete classification of rotationally symmetric self-shrinkers.

No direct analogue of Delaunay unduloids for self-shrinkers.

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 [Ang], 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.