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Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

Shanwei Ding, Jinyu Gao, Guanghan Li

TL;DR

This work develops a locally constrained, volume-preserving anisotropic mean curvature flow for star-shaped $\omega_0$-capillary hypersurfaces in the half-space and proves global existence with smooth convergence to an $\omega_0$-capillary Wulff shape. A novel inverse anisotropic Gauss map reparameterization provides curvature control and convexity preservation under anisotropic capillary boundary, enabling a rigorous flow-based route to sharp inequalities. The authors introduce general anisotropic capillary quermassintegrals $\mathcal V_{k,\omega_0}$ and prove their monotonicity, yielding new Alexandrov–Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces, with equality precisely for capillary Wulff shapes. Collectively, these results extend capillary and anisotropic convex geometry from static inequalities to a dynamic, flow-driven framework with explicit geometric characterizations.

Abstract

In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic $ω_0$-capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all $t>0$ by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the $ω_0$-capillary Wulff shape is a $θ$-capillary hypersurface with constant contact angle $θ$, the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two $θ$-capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the $ω_0$-capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two $θ$-capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry.

Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities

TL;DR

This work develops a locally constrained, volume-preserving anisotropic mean curvature flow for star-shaped -capillary hypersurfaces in the half-space and proves global existence with smooth convergence to an -capillary Wulff shape. A novel inverse anisotropic Gauss map reparameterization provides curvature control and convexity preservation under anisotropic capillary boundary, enabling a rigorous flow-based route to sharp inequalities. The authors introduce general anisotropic capillary quermassintegrals and prove their monotonicity, yielding new Alexandrov–Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces, with equality precisely for capillary Wulff shapes. Collectively, these results extend capillary and anisotropic convex geometry from static inequalities to a dynamic, flow-driven framework with explicit geometric characterizations.

Abstract

In this paper, an anisotropic volume-preserving mean curvature type flow for star-shaped anisotropic -capillary hypersurfaces in the half-space is studied, and the long-time existence and smooth convergence to a capillary Wulff shape are obtained. If the initial hypersurface is strictly convex, the solution of this flow remains to be strictly convex for all by adopting a new approach applicable to anisotropic capillary setting. In analogy with closed hypersurfaces, if the -capillary Wulff shape is a -capillary hypersurface with constant contact angle , the quermassintegrals for anisotropic capillary hypersurfaces match the mixed volume of two -capillary convex bodies. Thus, generalized quermassintegrals for anisotropic capillary hypersurfaces with general Wulff shapes (i.e., the -capillary Wulff shape has a variable contact angle) can be defined, which satisfy certain monotonicity properties along the flow. As applications, we establish an anisotropic capillary isoperimetric inequality for star-shaped anisotropic capillary hypersurfaces and a family of new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces. In particular, we provide a flow's method to derive the Alexandrov-Fenchel inequalities for two -capillary hypersurfaces, demonstrated in [30] (arXiv:2408.13655) from the view of point in convex geometry.
Paper Structure (19 sections, 39 theorems, 335 equations, 2 figures)

This paper contains 19 sections, 39 theorems, 335 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Wulff shape and dual Minkowski norm
  4. Anisotropic curvature
  5. New metric and anisotropic formulas
  6. Anisotropic support function and area element
  7. Proof of Theorem \ref{['thm:flow']}
  8. Evolution equations
  9. Radial graph parametrisation and the short time existence
  10. $C^0$ estimate
  11. $C^1$ estimates
  12. Long time existence and convergence
  13. Monotonic quantities and capillary isoperimetric inequality
  14. Proof of convexity preservation
  15. The uniform bound of $H_F$ along flow
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\omega_0 \in\left(-F\left(E_{n+1}\right), F\left(-E_{n+1}\right)\right)$. If the initial anisotropic $\omega_0$-capillary hypersurface $\Sigma_0$ is star-shaped with respect to the origin in $\overline{\mathbb{R}_{+}^{n+1}}$, then flow equ:flow has a unique smooth solution for all time. Moreove

Figures (2)

  • Figure 1: ${\mathcal{W}}$ and $\widetilde{{\mathcal{W}}}$
  • Figure 2: The relationship between $z$ and $\bar{z}$

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.2
  • Corollary 1.5
  • Remark 1.3
  • Conjecture 1.1
  • Lemma 2.1
  • ...and 66 more