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Parallel mean curvature surfaces with constant contact angle along free boundaries

Rui Gao, Miaomiao Zhu

TL;DR

The paper classifies branched immersed PMC surfaces in space forms under a constant non-zero boundary contact angle, establishing a complete codimension-two rigidity result and a codimension-reduction theorem for higher-genus bordered domains. The main technique combines a holomorphic Hopf-type quadratic differential framework with a dimension-reduction mechanism: first show the surface lies in a totally umbilic submanifold, then apply Schwarz reflection to force rigidity. In codimension two, the authors obtain a full classification of disks, describing when the image is a spherical cap, a plane disk, or a piece of a totally geodesic surface, with ambient type dictated by the curvature. For higher codimensions, they prove that the surface must either be minimal in a totally umbilic hypersurface or have constant mean curvature in a 3D totally umbilic submanifold, with boundary angles non-decreasing along the ambient reductions. The paper also furnishes explicit non-trivial examples of branched minimal immersions with non-orthogonal boundary contact angles to illustrate sharpness and the nontrivial interaction between angle conditions and codimension reduction.

Abstract

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result.

Parallel mean curvature surfaces with constant contact angle along free boundaries

TL;DR

The paper classifies branched immersed PMC surfaces in space forms under a constant non-zero boundary contact angle, establishing a complete codimension-two rigidity result and a codimension-reduction theorem for higher-genus bordered domains. The main technique combines a holomorphic Hopf-type quadratic differential framework with a dimension-reduction mechanism: first show the surface lies in a totally umbilic submanifold, then apply Schwarz reflection to force rigidity. In codimension two, the authors obtain a full classification of disks, describing when the image is a spherical cap, a plane disk, or a piece of a totally geodesic surface, with ambient type dictated by the curvature. For higher codimensions, they prove that the surface must either be minimal in a totally umbilic hypersurface or have constant mean curvature in a 3D totally umbilic submanifold, with boundary angles non-decreasing along the ambient reductions. The paper also furnishes explicit non-trivial examples of branched minimal immersions with non-orthogonal boundary contact angles to illustrate sharpness and the nontrivial interaction between angle conditions and codimension reduction.

Abstract

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result.
Paper Structure (16 sections, 6 theorems, 118 equations, 1 figure)

This paper contains 16 sections, 6 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main Results
  4. Some Explicit Examples
  5. An example in R4
  6. A generalization to Lg
  7. Veronese Minimal Immersion with Non-orthonormal Contact Angle in S4
  8. Difficulties and Main Ideas
  9. Organization of the paper
  10. Preliminaries and Construction of Quadratic Differentials
  11. Settings and Basic Properties
  12. Construction of Quadratic Differentials
  13. A Dimension Reduction Result
  14. Proof of Main Theorem
  15. Proof of Theorem \ref{['main theorem 1']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $u : D \rightarrow B^4$ be a branched immersion with non-zero parallel mean curvature vector from the unit disk $D \subset \mathbb R^2$ into a geodesic ball $B^4$ of a $4$-dimensional space form $\mathbb R^4(c)$ of constant curvature $c \in \mathbb R$ such that $u(D)$ meets $\partial B^4$ at a c Furthermore, if $u: A(r,R) \rightarrow B^4$ is a branched immersion with non-zero parallel mean cur

Figures (1)

  • Figure 1: The contact angle between $u(\Sigma)$ and $\mathcal{K}$ along the boundary $u(\partial \Sigma)$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: PMC branched immersion with constant contact angle
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 6 more