Starting point of the nice and well-organized paper under review are the so-called “symmetric Wente tori” in the Euclidean 2-space \(E^3\). These surfaces are closed (namely compact and without boundary), they have constant mean curvature and contain a continuous family of planar principal curves.
The author obtains various lower and upper bounds for the index of a symmetric Wente torus. He shows firstly, that the index is at least 7. Specially, the two simplest symmetric Wente Tori have index al least 9 and 8. He then establishes an algorithm for computing the index exactly. This algorithm is applied to give numerical estimations for the index of 17 different symmetric Wente tori. These estimations give with their turn rise to pose the conjecture that every symmetric Wente torus has an index at least 9.