Paper · May 19, 2020
Contracting self-similar solutions of nonhomogeneous curvature flows
As on arXiv
Abstract
A recent article by Li and Lv considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.
- Submitted
- May 19, 2020
- Subjects
- math.AP
Comments: 13 pages
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Read around it
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
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6C.8Monotonicity, ε-Regularity and Tangent Cones
Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.