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

Nicolau S. Aiex, Han Hong

TL;DR

The paper proves a compactness theorem for free boundary CMC surfaces in a compact 3-manifold with boundary under uniform geometric and topological bounds. It develops L^2-curvature control via Gauss–Bonnet, refines local curvature estimates through blow‑up arguments to obtain pointwise bounds, and establishes removable singularities for interior and boundary points. A key multiplicity analysis using weak stability and barrier constructions yields 1-sheeted convergence away from a finite singular set. Overall, the work extends Fraser–Li's minimal-surface compactness to the CMC free-boundary setting and removes the need for positive Ricci curvature, providing a robust framework for the moduli of free boundary CMC surfaces and their convergence properties.

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}.

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

TL;DR

The paper proves a compactness theorem for free boundary CMC surfaces in a compact 3-manifold with boundary under uniform geometric and topological bounds. It develops L^2-curvature control via Gauss–Bonnet, refines local curvature estimates through blow‑up arguments to obtain pointwise bounds, and establishes removable singularities for interior and boundary points. A key multiplicity analysis using weak stability and barrier constructions yields 1-sheeted convergence away from a finite singular set. Overall, the work extends Fraser–Li's minimal-surface compactness to the CMC free-boundary setting and removes the need for positive Ricci curvature, providing a robust framework for the moduli of free boundary CMC surfaces and their convergence properties.

Abstract

We prove that the space of free boundary CMC surfaces of bounded topology, bounded area and bounded boundary length is compact in the 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}.

Paper Structure

This paper contains 5 sections, 8 theorems, 50 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Curvature estimate
  4. Removable singularities
  5. Compactness Theorem

Key Result

Theorem \ref{compactness fbcmc}

Let $N$ be a compact $3$-dimensional manifold with boundary. Suppose $H_{\partial N}> H_0$ with respect to the inward conormal of $N$ along $\partial N$ and let $\Sigma_i$ be a sequence of connected embedded free boundary CMC surfaces with mean curvature $H_i$, genus $g_i$ and number of ends $r_i$ s Then there exists a smooth properly almost embedded CMC surface $\Sigma\subset N$ and a finite set

Figures (8)

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  • ...and 3 more figures

Theorems & Definitions (37)

  • Theorem \ref{compactness fbcmc}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Claim 2.2.1
  • Claim 2.2.2
  • Claim 2.2.3
  • Claim 2.2.4
  • Remark 2.3
  • Definition 2.4
  • ...and 27 more