Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary. (English) Zbl 1295.53062

Fraser, Ailana; Li, Martin Man-Chun

J. Differ. Geom. 96, No. 2, 183-200 (2014).

The main result in this paper is the following theorem: Let \(M^3\) be a compact 3-dimensional Riemannian manifold with nonempty boundary \(\partial M\). Suppose \(M\) has nonnegative Ricci curvature and the boundary \(\partial M\) is strictly convex with respect to the inward unit normal. Then, the space of compact properly embedded minimal surfaces of fixed topological type in \(M\) with free boundary \(\partial M\) is compact in the \(C^k\) topology for any \(k\geq 2\).

Reviewer: Huili Liu (Shenyang)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58D10	Spaces of embeddings and immersions

Keywords:

minimal surface; compactness; Ricci curvature; free boundary; topology