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Capillary hypersurfaces, Heintze-Karcher's inequality and Zermelo's navigation

Guofang Wang, Chao Xia

TL;DR

This work extends Heintze–Karcher-type inequalities to capillary hypersurfaces inside the unit ball by leveraging a Randers-type Finsler metric arising from Zermelo's navigation on the hyperbolic background. The authors develop a geodesic normal flow with respect to the Randers metric, translating capillary boundary conditions into free boundary problems in Finsler geometry and establishing a monotone deficit that yields the HK-type bound: ∫_Σ (x_{n+1}+cosθ_0⟨ν,E_{n+1}⟩)/H ≥ (n+1)/n ∫_Ω x_{n+1}, with equality characterizing θ_0-capillary spherical caps. The approach unifies capillary and free boundary geometry through navigation data (g_h, v_0) and shows the ball's convexity under the Finsler metric ensures flow surjectivity, while also offering a path to Reilly-type formulas in the Finsler setting. Extensions to the half-space are discussed, and the framework provides a versatile bridge between Riemannian and Finsler techniques for capillary boundary problems.

Abstract

In this paper, we establish a Heintze-Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect to this Finsler metric. Our results indicate that the relationship between capillary hypersufaces and hypersurfaces with free boundary is similar to the one between Finsler geometry and Riemannian geometry.

Capillary hypersurfaces, Heintze-Karcher's inequality and Zermelo's navigation

TL;DR

This work extends Heintze–Karcher-type inequalities to capillary hypersurfaces inside the unit ball by leveraging a Randers-type Finsler metric arising from Zermelo's navigation on the hyperbolic background. The authors develop a geodesic normal flow with respect to the Randers metric, translating capillary boundary conditions into free boundary problems in Finsler geometry and establishing a monotone deficit that yields the HK-type bound: ∫_Σ (x_{n+1}+cosθ_0⟨ν,E_{n+1}⟩)/H ≥ (n+1)/n ∫_Ω x_{n+1}, with equality characterizing θ_0-capillary spherical caps. The approach unifies capillary and free boundary geometry through navigation data (g_h, v_0) and shows the ball's convexity under the Finsler metric ensures flow surjectivity, while also offering a path to Reilly-type formulas in the Finsler setting. Extensions to the half-space are discussed, and the framework provides a versatile bridge between Riemannian and Finsler techniques for capillary boundary problems.

Abstract

In this paper, we establish a Heintze-Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect to this Finsler metric. Our results indicate that the relationship between capillary hypersufaces and hypersurfaces with free boundary is similar to the one between Finsler geometry and Riemannian geometry.
Paper Structure (7 sections, 16 theorems, 96 equations)

This paper contains 7 sections, 16 theorems, 96 equations.

Table of Contents

  1. introduction
  2. Free boundary hypersurfaces in Lg
  3. Zermelo's navigation and Finsler metrics
  4. Capillary hypersurfaces and Heintze-Karcher-type inequality
  5. Zermelo's navigation and capillary hypersurfaces in a ball
  6. Heintze-Karcher-type inequality in a ball
  7. Heintze-Karcher-type inequality in the half-space

Key Result

Theorem 1.1

Let $\theta_0\in (0, \pi)$ and ${\Sigma}\subset \overline{\mathbb{B}^{n+1}}\cap \mathbb{R}^{n+1}_+$ be a compact hypersurface with boundary $\partial{\Sigma}\subset \partial \mathbb{B}^{n+1}$ such that and $\partial{\Sigma}$ intersects $\partial \mathbb{B}^{n+1}$ transversally with contact angle $\theta(x)\le \theta_0$ for each $x\in \partial{\Sigma}$. Assume that ${\Sigma}$ is strictly mean conv

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 20 more