Free boundary constant mean curvature surfaces in a strictly convex three-manifold

TL;DR

This paper establishes conditions under which free boundary constant mean curvature surfaces in convex 3-manifolds are topologically simple or specific classical surfaces, extending understanding of their geometry and classification.

Contribution

It introduces a pinching condition on the traceless second fundamental form that classifies free boundary CMC surfaces in convex domains, including space forms.

Findings

Surfaces are homeomorphic to a disk or annulus under the pinching condition.

In space forms, surfaces are either spherical caps or Delaunay surfaces.

Provides geometric criteria for classifying free boundary CMC surfaces.

Abstract

Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $\Sigma$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $\Sigma$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $\Sigma$ is either a spherical cap or a Delaunay surface.