As on arXiv
Abstract
We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving hypersurfaces are assumed to stick tangentially to the boundary upon contact. In smooth bounded domains, we prove an existence and uniqueness theorem for weak solutions, and establish regularity of the level sets up to the obstacle. The proof combines various techniques, including elliptic regularization, blow-up analysis, and certain parabolic estimates. As an analytic application, we address the well-posedness problem for the usual weak inverse mean curvature flow, showing that the initial value problem always admits a unique maximal (or innermost) weak solution.
Comments: v3 update: expositional improvements and typo fixing. 72 pages, 6 figures. To appear in Geom. Topol
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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Infinity-harmonic functions and inverse mean curvature flow clusters
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Same author Kai Xu · Shared terms “obstacle”, “boundary condition”
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Topics Inverse mean curvature flow, Other curvature flows · Same author Kai Xu · Shared terms “weak inverse”
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Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Kai Xu
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Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact
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Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “level set”
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Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “level set”
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Topics Inverse mean curvature flow, Other curvature flows · Same author Kai Xu · Shared terms “weak inverse”
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On the evolution of hypersurfaces along their inverse spacetime mean curvature
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact
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
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 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.