Parallel mean curvature surfaces with constant contact angle along free boundaries

TL;DR

This paper classifies certain branched immersed disks with parallel mean curvature and constant contact angle boundaries in space forms, extends results to higher codimension, and provides explicit examples demonstrating the sharpness of these classifications.

Contribution

It introduces a classification of branched immersed disks with parallel mean curvature and constant contact angle, and proves a codimension reduction theorem for higher genus surfaces.

Findings

Classification of branched immersed disks with parallel mean curvature

Codimension reduction theorem for higher genus surfaces

Explicit examples of branched minimal immersions with free boundary conditions

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.