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

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 $$k\geq 2$$.

MSC:

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