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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.

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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.

enter image description here

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  • $\begingroup$ Thanks Sebastian! $\endgroup$
    – Leo Moos
    Commented Oct 16, 2021 at 16:20

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