Paper · Nov 9, 2026

A capillary logarithmic Gauss curvature flow and the capillary Minkowski problem

Xinqun Mei, Liangjun Weng

math.APWider flowsarXiv:2611.12455
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Abstract

We study a logarithmic Gauss curvature flow of strictly convex capillary hypersurfaces in the Euclidean half-space. For an acute contact angle and an arbitrary positive smooth prescribed function, we prove that a suitable initial dilation yields a global solution which converges smoothly after a horizontal translation with a uniquely determined constant velocity. As an application, we provide a logarithmic flow proof of the capillary Minkowski problem, previously established by Mei-Wang-Weng [Adv. Math. 469, 110230 (2025)].

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Submitted
Nov 9, 2026
Subjects
math.AP
Other curvature flowsGauss curvature flowFree boundary & capillaryLong-time existence & convergence

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  6. Capillary Orlicz-Minkowski flow in the upper half-space

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    Topics Gauss curvature flow, Free boundary & capillary, Long-time existence & convergence · Shared terms “half space”, “Minkowski”

  7. Flow by Gauss curvature to the -Gaussian Minkowski problem

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  8. The dual Orlicz-Minkowski problem for capillary hypersurfaces

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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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)