Self-shrinkers with a rotational symmetry

By S. Kleene, Niels Moller. 2010

100 citations recorded by the source corpus.

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.

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Citations in the corpus

Halfspace type theorems for self‐shrinkers (2014, methodology): “Our proof is geometrical and uses a catenoid type hypersurface discovered by Kleene– Moller [11].”
Unique asymptotics of ancient convex mean curvature flow solutions (2015, methodology): “The surfaces Σ̃b outside the cylinder were constructed by Kleene and Møller in [15].”
Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends (2011, methodology): “In [28], Kleene and Møller constructed a smooth embedded self-shrinking end of revolution in R that is asymptotic to C.”
Ancient asymptotically cylindrical flows and applications (2019, unknown): “1], see also [KM14] and [ADS19b, Sec.”
Complete self-shrinkers of the mean curvature flow (2012, background): “In [33], Kleene and Mφller proved that a complete embedded selfshrinking hypersurface of revolution is either a hyperplane R in R, a round cylinder R×Sn−1( √ n− 1), a round sphere S( √ n) or a smooth embedded torus.”