# A counterexample to a conjecture of Lawson

Yau quotes Lawson as having formulated the following conjecture [1]:

Let $$M$$ be an embedded minimal surface in $$\mathbf{S}^3$$. Prove that the two domains in $$\mathbf{S}^3$$ divided by $$M$$ have equal volume.

This is easily seen to hold for the simplest two minimal surfaces in $$\mathbf{S}^3$$, namely the equatorial spheres and the Clifford tori; it was also known at the time that the analogous statement fails in higher-dimensional spheres. (Whether it holds for the Lawson surfaces in $$\mathbf{S}^3$$ I do not know.)

Question. I remember reading a statement claiming that some surface had been found to contradict the proposed statement. Does somebody know which surface this is? Unfortunately I cannot seem to remember where I read this, and online searches are obscured by Lawson's other, more prominent conjecture.

[1] Shing-Tung Yau. Problem section. Seminar on Differential Geometry. 102 (1982) pp.669-706.

• math.tamu.edu/~jon.pitts/hp43/hp43.html I remember a talk, they found examples with extremely different volumes Commented Oct 16, 2021 at 0:21
• right, Pitts and Rubenstein 1988, comments on page 307 projecteuclid.org/journals/… They point out that, in their infinite set of examples, one may arrange one of the volumes to be arbitrarily small. Evidently an example is also in Karcher, Pinkall, Sterling Commented Oct 16, 2021 at 0:34
• projecteuclid.org/journals/journal-of-differential-geometry/… Commented Oct 16, 2021 at 0:43
• @WillJagy I forgot to say thank you to you as well; your comments are much appreciated! Commented Oct 16, 2021 at 21:05

## 1 Answer

As already pointed out by Will Jagy there is an example in this paper by Karcher-Pinkall-Sterling. It is build from a tetrahedral tesselation of $$S^3$$ with dihedral angles $$\tfrac{\pi}{2},$$ $$\tfrac{\pi}{2},$$ $$\tfrac{\pi}{2},$$ $$\tfrac{\pi}{3},$$ $$\tfrac{\pi}{2},$$ $$\tfrac{\pi}{5},$$ see the attached figure. The property of dividing the volumes into two equal parts holds for every Lawson surface $$\xi_{k,l}$$ since these admit a space reversing symmetry fixing the orientation of the surface.

• Thanks Sebastian! Commented Oct 16, 2021 at 16:20