As on arXiv
Abstract
We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.
Comments: 28 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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Higher regularity of the inverse anisotropic mean curvature flow
Topics Inverse mean curvature flow, Other curvature flows · Same authors Chaoqun Gao, Yong Wei, Rong Zhou · Shared terms “weak inverse”, “Huisken”, “Ilmanen”
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Positive -bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space
Topics Eigenvalues & spectrum · Same authors Yong Wei, Rong Zhou · Shared terms “hyperbolic space”, “endpoint”, “ingredient”
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On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
Topics Inverse mean curvature flow, Other curvature flows · Shared terms “minimizing domains”, “weak inverse”, “hyperbolic space”
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Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space
Topics Other curvature flows · Same authors Yong Wei, Rong Zhou · Shared terms “hyperbolic space”
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Alexandrov–Fenchel inequalities for convex domains in the sphere
Topics Other curvature flows · Same authors Yong Wei, Rong Zhou
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Contraction of hypersurfaces with positive sectional curvature in hyperbolic space
Topics Other curvature flows · Same authors Yong Wei, Rong Zhou · Shared terms “hyperbolic space”
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Asymptotic behaviour of the weak inverse anisotropic mean curvature flow
Topics Mean curvature flow · Same authors Chaoqun Gao, Yong Wei, Rong Zhou · Shared terms “weak inverse”, “Huisken”, “Ilmanen”
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A Minkowski type inequality in warped cylinders
Topics Inverse mean curvature flow, Other curvature flows · Shared terms “outward minimizing”, “hyperbolic space”
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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.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.