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A mean curvature type flow with capillary boundary in a horoball in hyperbolic space

Jinyu Guo

TL;DR

The paper studies a mean curvature type flow with θ-capillary boundary in a horoball of hyperbolic space, showing long-time existence and smooth convergence to an umbilical capillary hypersurface while preserving the enclosed volume. The authors reduce the flow to a scalar parabolic equation on a hemispherical domain via a radial graph parametrization, derive uniform a priori estimates, and employ barrier arguments with model umbilical surfaces to prove convergence to a unique limit around $E_{n+1}$. A weighted Minkowski formula in the horoball underpins the approach, and the energy functional monotonically decreases, yielding an isoperimetric-type inequality for capillary hypersurfaces in the horoball. The results extend Guan-Li-type flows to hyperbolic horoballs and provide a framework for energy-minimizing capillary hypersurfaces with prescribed volume, with special cases including free boundary and potential broad applications.

Abstract

In this paper, we study a mean curvature type flow with capillary boundary in a horoball in hyperbolic space. Our flow preserves the volume of the bounded domain enclosed by the hypersurface and monotonically decreases the energy functional. We show that it has the long time existence and converges to a truncated umbilical hypersurface in hyperbolic space. As an application, we solve an isoperimetric type problem for hypersurfaces with capillary boundary in a horoball.

A mean curvature type flow with capillary boundary in a horoball in hyperbolic space

TL;DR

The paper studies a mean curvature type flow with θ-capillary boundary in a horoball of hyperbolic space, showing long-time existence and smooth convergence to an umbilical capillary hypersurface while preserving the enclosed volume. The authors reduce the flow to a scalar parabolic equation on a hemispherical domain via a radial graph parametrization, derive uniform a priori estimates, and employ barrier arguments with model umbilical surfaces to prove convergence to a unique limit around . A weighted Minkowski formula in the horoball underpins the approach, and the energy functional monotonically decreases, yielding an isoperimetric-type inequality for capillary hypersurfaces in the horoball. The results extend Guan-Li-type flows to hyperbolic horoballs and provide a framework for energy-minimizing capillary hypersurfaces with prescribed volume, with special cases including free boundary and potential broad applications.

Abstract

In this paper, we study a mean curvature type flow with capillary boundary in a horoball in hyperbolic space. Our flow preserves the volume of the bounded domain enclosed by the hypersurface and monotonically decreases the energy functional. We show that it has the long time existence and converges to a truncated umbilical hypersurface in hyperbolic space. As an application, we solve an isoperimetric type problem for hypersurfaces with capillary boundary in a horoball.
Paper Structure (3 sections, 12 theorems, 106 equations, 1 figure)

This paper contains 3 sections, 12 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scalar flow
  3. A priori estimates and convergence

Key Result

Proposition 1.1

$(\rm i)$ The vector fields $x$ and $\{E_\alpha\}_{\alpha=1}^{n}$ are Killing vector fields in $\mathbb{H}^{n+1}$, i.e, $(\rm {ii})$$E_{n+1}$ is a conformal Killing vector field in $\mathbb{H}^{n+1}$, i.e, Here $\bar{\nabla}_i = \bar{\nabla} _{E_i}$ and $\bar{g}_{ij}= \bar{g}(E_i, E_j)$.

Figures (1)

  • Figure 1: Hypersurface $\Sigma$ with $\theta$-capillary boundary in the horoball $\mathcal{H}_{+}$.

Theorems & Definitions (22)

  • Proposition 1.1: GWX
  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3: GX2
  • Proposition 1.4: GX2
  • Definition 1.3
  • Theorem 1.1
  • ...and 12 more