As on arXiv
Abstract
This paper proves that, in mean curvature flow of a compact surface in a complete -manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact.
Comments: 15 pages
Its neighbourhood
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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Fattening in mean curvature flow
Topics Singularities · Same author Brian White · Shared terms “tangent flow”
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Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture
Topics Singularities · Same author Brian White
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Genus one singularities in mean curvature flow
Topics Singularities
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Generating Shrinkers by Mean Curvature Flow
Topics Singularities · Same author Brian White
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Graphical mean curvature flow with bounded bi-Ricci curvature
Topics Mean curvature flow · Shared terms “decreasing property”
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The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow
Topics Mean curvature flow · Same author Brian White
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Translating Annuli for Mean Curvature Flow
Topics Mean curvature flow · Same author Brian White
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The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators
Topics Mean curvature flow · Same author Brian White
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
- 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.