Existence of connected multi-ended exterior free-boundary minimal surfaces

Establish whether there exists a connected exterior free-boundary minimal surface in \(\mathbb R^3\setminus B^3\) with compact boundary and at least two regular ends.

Background

The paper proves a spectral discreteness theorem for exterior free-boundary minimal surfaces with any finite number of regular ends, including the multi-ended case. However, the geometric existence of connected examples with two or more regular ends is not established. The authors explicitly note that no such connected surface is known, while disjoint unions provide only trivial examples.

References

No connected exterior FBMS in $\mathbb R3$ with $L\geq2$ regular ends is known (disjoint unions, such as $C_\alpha$ together with its mirror image in ${z=0}$, trivially have $L=2$); Theorem~\ref{thm:multiple_ends} describes connected ones should they exist.

— Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature  (2609.29318 - Kozyrev, 24 Sep 2026) in Section 5, “Several Ends”