Self-shrinkers with a rotational symmetry

TY - JOUR

T1 - Self-shrinkers with a rotational symmetry

AU - Kleene, Stephen J.

AU - Møller, Niels Martin

PY - 2014

Y1 - 2014

N2 - In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σn ⊆ ℝn+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in ℝn+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 Σn is either a hyperplane ℝn, the round cylinder ℝ × Sn−1 of radius (Formula Presented), the round sphere Sn of radius (Formula Presented), or is diffeomorphic to an S1 × Sn−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.

AB - In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σn ⊆ ℝn+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in ℝn+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 Σn is either a hyperplane ℝn, the round cylinder ℝ × Sn−1 of radius (Formula Presented), the round sphere Sn of radius (Formula Presented), or is diffeomorphic to an S1 × Sn−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.

M3 - Journal article

SN - 0002-9947

VL - 366

SP - 3943

EP - 3963

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -