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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.

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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

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  • $\begingroup$ That's a fine explanation, thanks! Is there a simple way to see that the non-orthogonal Scherk surfaces must be offset by a positive amount? $\endgroup$
    – Leo Moos
    Oct 28 '20 at 20:57
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    $\begingroup$ @LeoMoos I'm not sure if there is a simple way. It's possible you could quotient out by the period translation and see if something about the flux or torque of the end of the resulting surface gave it to you, but I don't know if that works off the top of my head. Alternatively, you could observe that as the wings of the Scherk family collapse to one another, an appropriate rescaling gives the catenoid and this is logarithmicaly offset from the tangent plane. $\endgroup$
    – RBega2
    Oct 28 '20 at 22:29
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Let me at some additional remarks to RBega2s excellent answer. As pointed out by Otis Chodosh in his comment, the Lawson surfaces $\xi_{1,g}$ are isolated by a result of Kapouleas and Wiygul. Therefore, it is not possible to desingularies two intersecting minimal 2-spheres meeting at an angle $\theta\neq\tfrac{\pi}{2}$, by replacing the circle of intersection with a singly periodic Scherk surface of angle $\theta.$

On the other hand, by combining the result of Kapouleas-Wiygul with the work of Kusner-Mazzeo-Pollack about 'The moduli space of complete embedded constant mean curvature surfaces', the Lawson surface $\xi_{1,g}$ can be deformed through a (local) family of ConstantMeanCurvature surfaces in the 3-sphere. For $g=1$, this family is well-known, and can be described explicitly: it deforms the Clifford torus ($=\xi_{1,1}$) through homogenous CMC tori and then bifurcates off to 2-lobed Delaunay tori which degenerate to a minimal 2-sphere of multiplicity 2. In a certain sense (as described below) these CMC surfaces converge against 2 non-perpendicularly intersecting minimal 2-spheres.

With Lynn Heller and Martin Traizet, we have shown in a recent preprint that, at least for large genus $g$, there are complete families of embedded CMC surfaces $\Gamma_{g,\varphi}$, parametrized by $\varphi\in(0,\tfrac{\pi}{2})$ (this angle determines the conformal type of the surface $\Gamma_{g,\varphi}$), such that

  1. For $g\to \infty$ and fixed $\varphi\in(0,\tfrac{\pi}{2})$, the CMC surfaces converge against 2 intersecting minimal 2-spheres which intersect with angle $2\varphi.$ The blowup along the circle of intersection is the singly periodic Scherk surface of angle $2\varphi$.

  2. The CMC surface $\Gamma_{g,\varphi}$ is minimal (for large $g$) if and only if $\varphi=\frac{\pi}{4}$. These are exactly the Lawson surfaces $\Gamma_{g,\tfrac{\pi}{4}}=\xi_{1,g}.$

  3. For fixed $g$ large, the surfaces $\Gamma_{g,\varphi}$ converge against the minimal 2-sphere of multiplicity 2 for $\varphi\to0,\tfrac{\pi}{2}.$ This means the families are complete.

  4. The mean curvature $H$ of $\Gamma_{g,\varphi}$ is given via a smooth function $\tilde H$ as $H(g,\varphi)=\tilde H(\tfrac{1}{2g+2},\varphi)$ such that $\tilde H(0,\varphi)=0$ and $\frac{d}{dt}_{\mid t=0} \tilde H(t,\varphi)=-2 \sin(2\varphi)\log(\tan(\varphi)).$

It should be possible to interpret the last formula as a quantitative statement about the translational defect of the wings, since the four minimal hemi-spheres need to be replaced by parallel surfaces wich are (halfs of) round CMC spheres.

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    $\begingroup$ I appreciate the added information. Congratulations on the preprint - it seems like a very interesting result! By the way, if I may add some questions: what does it means for a family of surfaces to be complete? Also, if I am not mistaken the deformations of the Clifford torus basically foliate the sphere. I am guessing this isn't true anymore here? Do your methods explain how $H$ changes with the angle $\varphi$? $\endgroup$
    – Leo Moos
    Aug 24 at 17:30
  • $\begingroup$ Many thanks. Completeness of the family of surfaces means, that the family of induced Riemann surface structures converges to points in the boundary of the Teichmüller space at $\varphi=0,\tfrac{\pi}{2}.$ In particular, the family cannot be extended any further, and has a nice (degenerate) geometric limit by point (3.). I do not know if this family foliates the 3-sphere. Concerning the mean curvature: (4.) describes the dependence on $\varphi$ for $g\sim\infty$. It is possible to compute $H$ to arbitrary order (in $t=\frac{1}{2g+2}$) up to certain complicated integrals (multipolylogs) $\endgroup$
    – Sebastian
    Aug 24 at 18:19
  • $\begingroup$ That's very neat, thanks again! $\endgroup$
    – Leo Moos
    Aug 24 at 22:23

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