Mean curvature self-shrinkers of high genus: Non-compact examples
| Autor: | Kapouleas, Nikolaos, Kleene, Stephen J., Møller, Niels Martin |
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| Rok vydání: | 2011 |
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| Zdroj: | J. Reine Angew. Math. 739 (2018), 1--39. MR3808256 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1515/crelle-2015-0050 |
| Popis: | 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 H\"older spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics. Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew. Math |
| Databáze: | arXiv |
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