The Lawson minimal surfaces $\xi_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [Law70]
these were constructed from geodesic triangulations. An alternative construction was given by Kapouleas [Kap10]. This is obtained by desingularising two orthogonal spheres $S_1,S_2 \subset \mathbf{S}^3$. These intersect along a great circle $C$, which roughly speaking is replaced with a 'bent copy' of the singly-periodic Scherk surface. Thus, away from a tubular neighbourhood of $C$ the Lawson surfaces $\xi_{1,g}$ have four connected components, called *wings* by Kapouleas, two of which near either $S_1,S_2$.

Kapouleas [Kap10,Question 4.3] then asks whether the Lawson surfaces can 'flap their wings': is it true that any minimal surface with four 'wings' away from the tubular neighbourhood of a great circle $C$ must be a Lawson surface? This is a question about the rigidity of the construction: two non-orthogonal spheres cannot be desingularised.

The question might look surprising, because there exists a one-parameter family of Scherk surfaces $\mathcal{S}_\theta \subset \mathbf{R}^3$, each asymptotic to a pair of planes meeting at an angle $\theta \in (0,\pi/2]$. One might thus naively assume that to desingularise two spheres $S_1,S_2 \subset \mathbf{S}^3$ forming an angle $\theta \in (0,\pi/2)$ one could plug in a 'bent' copy of $\mathcal{S}_\theta$, but Kapouleas' question points to an obstruction.

**Question**: What is the heuristic behind this obstruction? Why can two non-orthogonal spheres in $\mathbf{S}^3$ not be desingularised minimally, but two non-orthogonal planes in $\mathbf{R}^3$ can?

Some elements of an answer are given in [Kap10], but I am not sure I quite follow the explanation, in particular why the situation is so much more rigid in the sphere than Euclidean space.