Compactness of Constant Mean Curvature Surfaces in Three Manifold with Positive Ricci Curvature

TL;DR

This paper establishes a compactness theorem for constant mean curvature surfaces in three-manifolds with positive Ricci curvature, providing bounds on geometric properties and eigenvalues.

Contribution

It introduces a compactness theorem for constant mean curvature surfaces with bounds in three-manifolds with positive Ricci curvature, and derives eigenvalue estimates.

Findings

Proved a compactness theorem for such surfaces.

Established a lower bound for the first eigenvalue.

Applied the theorem to specific geometric contexts.

Abstract

In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.