Mean curvature self-shrinkers of high genus: Non-compact examples
By N. Kapouleas, S. Kleene, Niels Moller. Journal für die Reine und Angewandte Mathematik, 2011
94 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/826b6a6b1ecee3bc322ebada0a73f6a152a16f76
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc (2017, background): “ich can be used to substantially simplify the construction and proof (see 4.4). This is the case also for some recent desingularization constructions of self-shrinkers of the Mean Curvature Flow [44] [24] which are also based on [30]. The article is self-contained and we have carefully simpied the c”