Mean curvature self-shrinkers of high genus: Non-compact examples

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 4 ⁢ g + 4 {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 ℝ 3 {\mathbb{R}^{3}} over a 2 ⁢ π / ( g + 1 ) {2\pi/(g+1)} -periodic graph on an equator of the unit sphere 𝕊 2 ⊆ ℝ 3 {\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.