Author:
Kapouleas Nikolaos,Kleene Stephen James,Møller Niels Martin
Abstract
Abstract
We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent’s torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has
{4g+4}
symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry.
Each is at infinity asymptotic to the cone in
{\mathbb{R}^{3}}
over a
{2\pi/(g+1)}
-periodic graph on an equator of the unit sphere
{\mathbb{S}^{2}\subseteq\mathbb{R}^{3}}
, with the shape of a periodically “wobbling sheet”. This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions.
The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein–Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.
Funder
National Science Foundation
Subject
Applied Mathematics,General Mathematics
Reference58 articles.
1. Shrinking doughnuts;Nonlinear diffusion equations and their equilibrium states,1992
2. Constant mean curvature surfaces by fusing Wente tori;Invent. Math.,1995
3. Generic mean curvature flow II; dynamics of a closed smooth singularity;Preprint
4. Generation of analytic semigroups by elliptic operators with unbounded coefficients;SIAM J. Math. Anal.,1987
Cited by
34 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献