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

Abstract Let M be a compact three-dimensional Riemannian manifold with non-negative Ricci curvature and a non-empty boundary $\partial M$ . Fraser and Li [2] 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.