Isothermic Tori With One Family of Planar Curvature Lines and Area Constrained Hyperbolic Elastica

Isothermic surfaces form an integrable system and generalize constant mean curvature surfaces. Constant mean curvature tori are countable and classified using integrability into an integer hierarchy. The simplest are Wente tori, which have one family of planar curvature lines. Here, we begin a similar classification for isothermic tori. We classify isothermic tori with one family of planar curvature lines. These tori have functional freedom, which is unusual for integrable geometries. We give theta function formulas for the planar curves and prove they are area constrained hyperbolic elastica. Our results complete Darboux’s 1883 local classification using complex analytic methods. His choice of real reduction has no tori. We also characterize the finite dimensional space of isothermic tori with planar and spherical curvature lines, generalizing Wente tori.