Paper · Nov 25, 2022
Convexity of 2-convex translating and expanding solitons to the mean curvature flow in
As on arXiv
Abstract
In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdziński [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in . More precisely, for , we show that any -dimensional complete 2-convex translating solitons are convex, and any -dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.
Comments: 15 pages
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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.