Compactness of the Space of Free Boundary CMC Surfaces with Bounded Topology

TL;DR

This paper proves the compactness of the space of free boundary constant mean curvature surfaces with bounded topology, area, and boundary length, extending minimal surface results to the CMC setting.

Contribution

It establishes a CMC analogue of Fraser-Li's minimal surface compactness theorem, showing the space is compact in the $C^k$ sense away from finitely many points.

Findings

The space of free boundary CMC surfaces with bounded parameters is compact.

The result extends minimal surface theory to constant mean curvature surfaces.

Compactness holds away from a finite set of points.

Abstract

We prove that the space of free boundary CMC surfaces of bounded topology, bounded area and bounded boundary length is compact in the $C^k$ graphical sense away from a finite set of points. This is a CMC version of a result for minimal surfaces by Fraser-Li \cite{fraser.a-li.m2014}.