Mean Curvature Self-Shrinkers of High Genus: Non-Compact Examples

TY - JOUR

T1 - Mean Curvature Self-Shrinkers of High Genus: Non-Compact Examples

AU - Kapouleas, Nikolaos

AU - Kleene, Stephen J.

AU - Møller, Niels Martin

PY - 2015

Y1 - 2015

N2 - 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 ℝ3 over a 2π/(g + 1)-periodic graph on an equator of the unit sphere S2 ⊆ ℝ3, 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.

AB - 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 ℝ3 over a 2π/(g + 1)-periodic graph on an equator of the unit sphere S2 ⊆ ℝ3, 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.

U2 - 10.1515/crelle-2015-0050

DO - 10.1515/crelle-2015-0050

M3 - Journal article

SN - 0075-4102

VL - 2018

JO - Journal fuer die Reine und Angewandte Mathematik

JF - Journal fuer die Reine und Angewandte Mathematik

IS - 739

ER -