Embedded isothermic tori from holomorphic maps

Soap bubbles and soap films that span curved wireframes can be mathematically described using so-called minimal surfaces or surfaces of constant mean curvature (CMC surfaces). These surfaces appear not only in everyday physics but also in more abstract ambient spaces with constant curvature, such as in hyperbolic or elliptic space. All these surfaces belong to the wider class of isothermic surfaces. They are characterized by the local existence of conformal curvature linesin other words, these surfaces can be covered along their curvature lines by infinitesimal conformal squares. Isothermic surfaces were already studied by classical geometers such as Darboux and Bianchi. In more recent times, major progress has been made through the connection with integrable systems. However, many results in this field are only local in nature, meaning they only apply to small surface patches. Global propertiessuch as symmetries, periodicity, closedness, or the question of whether a surface can be embedded into space without self-intersectionshave only recently come into focus. So far, these global aspects are only partially understood. In our research project, we aim to investigate such global isothermic structures on tori, that is, surfaces that are topologically shaped like a doughnut. In particular, we focus on a special class of these tori that are covered by a family of spherical curvature lines. This class includes, for example, the famous Wente tori, the first known examples of closed, compact, non-round surfaces with constant mean curvature. A central tool in our project is a novel construction method called lifted-folding. This concept has been recently introduced in the context of discrete isothermic surfaces and allows such surfaces to be constructed from simple data: special holomorphic maps with circular folding axes and real folding functions. An advantage of this approach is that it facilitates control over global properties; the periodicity and embeddedness of each spherical curvature line is already determined by the choice of the holomorphic map.