Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

Xiaoxiang Chai, Yimin Chen

TL;DR

This work develops a constrained mean curvature flow for capillary hypersurfaces in hyperbolic space $\mathbb H^{n+1}$ supported on a totally geodesic plane, preserving the enclosed volume. It proves a new Minkowski type integral formula for such capillary hypersurfaces, enabling a volume-preserving flow whose scalar equation is derived via a radial function $u$ on $\mathbb S^n_+$; under a sharp angle condition, the flow exists globally and converges to a $\theta$-umbilical cap. Moreover, the authors show that a $\theta$-umbilical cap minimizes the capillary energy $\mathcal Q(\Sigma) = |\Sigma| - \cos\theta |\widehat{\partial\Sigma}|$ among caps with the same enclosed volume. Collectively, these results provide a variational perspective and stability for capillary profiles in hyperbolic space, with implications for geometric inequalities and constructive approaches to energy-minimizing capillary caps.

Abstract

We prove a new Minkowski type formula for capillary hypersurfaces supported on totally geodesic hyperplanes in hyperbolic space. It leads to a volume-preserving flow starting from a star-shaped initial hypersurface. We prove the long-time existence of the flow and its uniform convergence to a $θ$-totally umbilical cap. Additionally, we establish that a $θ$-totally umbilical cap is an energy minimizer for a given enclosed volume.

A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane

TL;DR

This work develops a constrained mean curvature flow for capillary hypersurfaces in hyperbolic space supported on a totally geodesic plane, preserving the enclosed volume. It proves a new Minkowski type integral formula for such capillary hypersurfaces, enabling a volume-preserving flow whose scalar equation is derived via a radial function on ; under a sharp angle condition, the flow exists globally and converges to a -umbilical cap. Moreover, the authors show that a -umbilical cap minimizes the capillary energy among caps with the same enclosed volume. Collectively, these results provide a variational perspective and stability for capillary profiles in hyperbolic space, with implications for geometric inequalities and constructive approaches to energy-minimizing capillary caps.

Abstract

We prove a new Minkowski type formula for capillary hypersurfaces supported on totally geodesic hyperplanes in hyperbolic space. It leads to a volume-preserving flow starting from a star-shaped initial hypersurface. We prove the long-time existence of the flow and its uniform convergence to a -totally umbilical cap. Additionally, we establish that a -totally umbilical cap is an energy minimizer for a given enclosed volume.
Paper Structure (7 sections, 9 theorems, 144 equations, 1 figure)

This paper contains 7 sections, 9 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski type formula
  4. Scalar equation of the flow
  5. $C^0$ estimate
  6. $C^1$-estimate
  7. Convergence of the flow

Key Result

Theorem 1

Let $x_0 : M \rightarrow P_+ \subset \mathbb{H}^{n + 1}$ be an embedding of a compact capillary hypersurface $\Sigma_0 = x_0 (M)$, supported on the totally geodesic plane $P$ with constant contact angle $\theta$. Suppose there exist constants c, R such that $K_0 (c, R, \theta) > c (n - 1) / 4 n$, a Then the flow flow exists globally with uniform $C^{\infty}$-estimates. Moreover, $x (\cdot, t)$ un

Figures (1)

  • Figure 1: $\mathcal{C}_1$, equidistant hypersurface; $\mathcal{C}_2$, geodesic ball; $\mathcal{C}_3$, horosphere.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • proof
  • Remark 3
  • ...and 11 more