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Quantitative Alexandrov theorem for capillary hypersurfaces in the half-space

Xiaohan Jia, Xuwen Zhang

TL;DR

This work provides a quantitative version of Alexandrov’s theorem for θ-capillary hypersurfaces in the half-space, showing that a capillary CMC surface with small $L^n$ deficit from a constant $\lambda$ is close to a finite collection of θ-caps and balls of radius $R=n/\lambda$ in the anisotropic half-space geometry. The authors integrate a Capillary–anisotropic reformulation via the capillary gauge $F_θ$, a capillary Michael–Simon inequality, and a Topping-type bound, establishing robust a priori estimates, a density control, and a shifted-distance comparison that culminates in explicit deficit–to–distance stability estimates. The main result extends Julin–Niinikoski’s quantitative soap bubble stability to capillary boundaries, with detailed height and mutual-distance controls for the centers and a precise characterization of equality cases as unions of θ-caps or θ-balls. The approach has potential implications for capillary mean curvature flow and the geometric structure of capillary interfaces, linking variational stability to explicit geometric configurations in the half-space.

Abstract

In this paper, we prove the quantitative version of the Alexandrov theorem for capillary hypersurfaces in the half-space, which generalizes Julin-Niinikoski's result to the capillary case. The proof is based on the quantitative analysis of the Montiel-Ros-type argument carried out in our joint works with Wang-Xia.

Quantitative Alexandrov theorem for capillary hypersurfaces in the half-space

TL;DR

This work provides a quantitative version of Alexandrov’s theorem for θ-capillary hypersurfaces in the half-space, showing that a capillary CMC surface with small deficit from a constant is close to a finite collection of θ-caps and balls of radius in the anisotropic half-space geometry. The authors integrate a Capillary–anisotropic reformulation via the capillary gauge , a capillary Michael–Simon inequality, and a Topping-type bound, establishing robust a priori estimates, a density control, and a shifted-distance comparison that culminates in explicit deficit–to–distance stability estimates. The main result extends Julin–Niinikoski’s quantitative soap bubble stability to capillary boundaries, with detailed height and mutual-distance controls for the centers and a precise characterization of equality cases as unions of θ-caps or θ-balls. The approach has potential implications for capillary mean curvature flow and the geometric structure of capillary interfaces, linking variational stability to explicit geometric configurations in the half-space.

Abstract

In this paper, we prove the quantitative version of the Alexandrov theorem for capillary hypersurfaces in the half-space, which generalizes Julin-Niinikoski's result to the capillary case. The proof is based on the quantitative analysis of the Montiel-Ros-type argument carried out in our joint works with Wang-Xia.
Paper Structure (23 sections, 16 theorems, 307 equations, 1 figure)

This paper contains 23 sections, 16 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantitative soap bubble problem
  3. Capillary Alexandrov's theorem
  4. Main result
  5. Capillarity and anisotropy
  6. Outline of the proof
  7. Acknowledgements
  8. Notations and preliminaries
  9. Notations
  10. Capillary hypersurfaces in the half-space
  11. Capillary gauge
  12. Capillary geometry and anisotropic geometry
  13. More on capillarity
  14. Michael-Simon-type inequality and Topping-type inequality
  15. A priori estimates
  16. ...and 8 more sections

Key Result

Theorem 1.1

Given $n\in\mathbb{N}^+$, $\theta\in(0,\pi)$, let ${\Sigma}\subset\overline{\mathbf{R}^{n+1}_+}$ be a compact, embedded $\theta$-capillary hypersurface. Let $\Omega$ denote the enclosed region of ${\Sigma}$ with $\partial\mathbf{R}^{n+1}_+$ such that $\partial\Omega={\Sigma}\cup T$. Given $\lambda\i and points such that if $P(\Omega)\leq C_0$, $\lvert\Omega\rvert\geq C_0^{-1}$ and then for $S_\t

Figures (1)

  • Figure 1: shifted distance function

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4: Trace estimate
  • proof
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • ...and 21 more