The Lawson minimal surfaces $\xi\_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [[Law70]](https://www.jstor.org/stable/1970625?seq=1)
these were constructed from geodesic triangulations. An alternative construction was given by Kapouleas [[Kap10]](https://arxiv.org/abs/1012.5788). 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](https://en.wikipedia.org/wiki/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]](https://arxiv.org/abs/1012.5788) 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]](https://arxiv.org/abs/1012.5788), 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.



Flapping wings: on a question of Kapouleas
My understanding of Kapouleas' heuristic is as follows.


In $\mathbb{R}^3$, a member Scherk family has four asymptotic half-planes. When you blow these down you get two intersecting planes (at any angle in $\theta\in (0, \pi/2]$) -- this union is the tangent cone at infinity. However, if you look at them in the original surface, they are offset from the tangent cone by a translation that depends on the angle $\theta$. It turns out that this translation factor is zero when $\theta=\frac{\pi}{2}$ i.e. when the wings are orthogonal (this can be seen by symmetry for instance). However, it is not zero for smaller angles. Basically, for orthogonal planes, the error for the asymptotic approximation of the Scherk surface by its tangent cone is $o(1)$ while for other angles it is $O(1)$.


This means that if you were trying to construct the Scherk surface by "gluing" you would need the flexibility to not only flap the wings but also to translate them slightly.


In the sphere we replace the wings with great hemi-spheres. The issue is that while we can still flap them, we can't translate them as parallel surfaces are no longer minimal. In other words, there is less flexibility and so one has to be in a situation where this is not an issue -- it turns out orthogonal wings is such a situation but others do not appear to be.


There may be other issues at play, but that is what I believe the gluing perspective of Kapouleas tells you


