As on arXiv
Abstract
We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding-Minicozzi respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds.
Comments: 27 pages, comments welcome
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Quenching for axisymmetric hypersurfaces under forced mean curvature flows
Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “forcing term”
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A stable self-shrinking Möbius bundle in
Topics Self-shrinkers & translators, Singularities, Stability & uniqueness · Same author Jonathan J. Zhu
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A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting
Topics Volume-preserving & constrained flows · Shared terms “forced mean”
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Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers
Topics Self-shrinkers & translators · Shared terms “Colding Minicozzi”, “closed self”, “Euclidean space”
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Mean curvature flow with generic initial data II
Topics Self-shrinkers & translators, Singularities · Shared terms “cylindrical self”
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Cylindrical tangent flows in mean curvature flow
Topics Singularities, Stability & uniqueness · Shared terms “Colding Minicozzi”
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Level-set forced mean curvature flow with the Neumann boundary condition
Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “forcing term”
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Uniqueness of tangent flows in free boundary flow
Topics Self-shrinkers & translators, Singularities, Stability & uniqueness · Shared terms “cylindrical self”
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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
- 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.
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.