As on arXiv
Abstract
In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.
Comments: 46 pages
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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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Uniqueness of capillary Gauss solitons
Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows · Same authors Xinqun Mei, Guofang Wang, Liangjun Weng · Shared terms “capillary hypersurfaces”
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Topics Free boundary & capillary, Other curvature flows · Same authors Xinqun Mei, Liangjun Weng · Shared terms “capillary hypersurfaces”, “counterpart”
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Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary
Topics Free boundary & capillary, Other curvature flows · Same authors Guofang Wang, Liangjun Weng · Shared terms “capillary hypersurfaces”, “flow converges”
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Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space
Topics Free boundary & capillary, Other curvature flows · Same authors Xinqun Mei, Liangjun Weng · Shared terms “capillary hypersurfaces”, “Guan”
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A constrained mean curvature flow and Alexandrov-Fenchel inequalities
Topics Free boundary & capillary · Same authors Xinqun Mei, Guofang Wang, Liangjun Weng
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Capillary Orlicz-Minkowski flow in the upper half-space
Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows
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Free boundary flows by powers of the Gauss curvature in the unit ball
Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows · Shared terms “normalized flow”, “Tso”, “finite time”
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Quermassintegral inequalities for capillary hypersurfaces in a horoball
Topics Free boundary & capillary, Other curvature flows · Same author Liangjun Weng · Shared terms “capillary hypersurfaces”
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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
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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
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Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.