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

TL;DR

This paper addresses the sharpness of Fraser–Li's free-boundary compactness theorem by showing that the strict convexity of the boundary cannot be relaxed. It constructs an explicit noncompact family of free-boundary minimal annuli in $(S^2_+ \times S^1,g)$ with $g$ the standard product metric, where $\partial(S^2_+ \times S^1)$ is totally geodesic. The authors derive a one-parameter family of free-boundary minimal annuli via a profile $r(t)$ solving $\frac{r''}{1+r'^2}=\cot r$, with free-boundary conditions leading to elliptic-function solutions; these yield a sequence with no convergent subsequence, proving noncompactness. The result highlights the necessity of the strict convexity condition in Fraser–Li's theorem and sharpens understanding of free-boundary minimal surface compactness in nonnegative Ricci manifolds.

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.

Theorems & Definitions (2)

• Theorem 1.1: Fraser-Li
• Theorem 1.2