Free boundary minimal annuli in $S^2_+\times S^1$

TL;DR

This paper investigates free boundary minimal annuli in a specific geometric setting, demonstrating that the previously assumed strict convexity condition on the boundary cannot be weakened.

Contribution

It proves that the strict convexity condition on the boundary is essential for the compactness theorem of minimal surfaces with free boundary.

Findings

Strict convexity of the boundary is necessary for the compactness theorem.

The relaxation of convexity condition invalidates the previous compactness results.

Abstract

Let $M$ be a compact 3-dimensional Riemannian manifold with nonnegative Ricci curvature and a nonempty boundary $\partial M$. Fraser and Li \cite{Fraser&Li} established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in $M$ with a free boundary on $\partial M$, assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.