Paper · Jul 11, 2023
Conformal solitons for the mean curvature flow in hyperbolic space
As on arXiv
Abstract
In this paper we study conformal solitons for the mean curvature flow in hyperbolic space . Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field . We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of . We conclude by proving rigidity results for bowl and grim-reaper cylinders.
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.