We consider the eigenvalue problem ΔS2ξ+2ξ=0 in
Ω and ξ=0 along ∂Ω, being Ω the
complement of a disjoint and finite union of smooth and bounded simply
connected regions in the two-sphere S2. Imposing that ∣∇ξ∣ is locally constant along ∂Ω and that ξ has infinitely
many maximum points, we are able to classify positive solutions as the
rotationally symmetric ones. As a consequence, we obtain a characterization of
the critical catenoid as the only embedded free boundary minimal annulus in the
unit ball whose support function has infinitely many critical points