Compact Bonnet Pairs: isometric tori with the same curvatures

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. We use the relationship between Bonnet pairs and isothermic surfaces. As a part of our construction we give a complete classification of isothermic surfaces with one family of planar curvature lines and of the corresponding cylinders.