Unrestricted boundary-end estimate under mean-convexity

Determine whether the boundary-end index estimate can be established for arbitrary boundary ends of complete two-sided free boundary minimal surfaces in smooth mean-convex domains of Euclidean three-space, without imposing the additional growth or capacity conditions used for selected boundary ends.

Background

The paper proves index estimates involving interior ends and selected boundary ends. For selected boundary ends, the argument requires a quantitative metric-growth condition, or more generally a capacity condition, to construct cutoff functions whose energy errors vanish when meromorphic harmonic one-forms with prescribed poles are used as test objects.

The authors explicitly identify as unresolved the extension of this estimate to arbitrary boundary ends under mean-convexity alone. Such an extension would remove the need to assume admissible pole orders or corresponding growth conditions at the boundary ends and would yield an unrestricted boundary-end estimate.