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 Σ n ⊆ R n+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in 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 Σ n is either a hyperplane R n , the round cylinder R × S n−1 of radius 2(n − 1), the round sphere S n of radius √ 2n, or is diffeomorphic to an S 1 × 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.

A Survey of Closed Self-Shrinkers with Symmetry

“…• Immersed self-shrinkers: Building on the work in [9,16,43], infinitely many immersed and non-embedded self-shrinkers for each of the rotational topological types: S n , S 1 × S n−1 , R n , and S 1 × R n−1 were constructed in [20]. The main idea for the construction is to study the behavior of solutions to the geodesic equation near two known self-shrinkers and use continuity arguments to find complete self-shrinkers between them.…”

“…The embeddedness assumption is necessary due to the existence of immersed and non-embedded S 3 self-shinkers constructed in [16,20]. Uniqueness is known in the rotational case [17,43].…”

Geometric Properties of Self-Shrinkers in Cylinder Shrinking Ricci Solitons

“…Recently, Cavalcante and Espinar [1] proved a halfspace theorem for self-shrinkers properly immersed in R n+1 . The proof uses the idea of Hoffman and Meeks and a catenoid type self-shrinker discovered by Kleene and Moller [20]. Note that there are many examples of compact self-shrinkers and noncompact proper self-shrinkers.…”

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

“…See f.ex. the halfspace theorem for self-shrinkers in [CE16], and note how the asymptotically conical self-shrinkers in [KM14] can easily be cut to get such examples which are noncompact with compact boundary.…”