Embedded isothermic tori from holomorphic maps

• Applicant: Dr. Gudrun Szewieczek
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552648860

Isothermic surfaces form a rich integrable class, including minimal and constant mean curvature (CMC) surfaces in space forms, and therefore play a central role in differential geometry. Away from umbilics, they are characterized by the existence of conformal curvature line coordinates. Most of the extensive theory on them is of a local nature. However, some global examples have led to groundbreaking discoveries: first and foremost Wente's famous CMC-torus. More recently, certain isothermic tori laid the foundations for finding compact Bonnet pairs, and the existence of minimal and CMC-solutions to free boundary and capillary problems was proved. All these global examples listed above have in common that they are foliated by a family of planar or spherical lines of curvature. This additional property of curvature lines is predestined for the construction of global isothermic surfaces: as we proved in [Cho-Pember-Szewieczek (2023)], such curvature lines are Möbius transforms of constrained elastic curves. Since these are either closed or quasi-periodic, isothermic surfaces with a family of spherical curvature lines are already globally defined in one coordinate direction. A general description for smooth isothermic surfaces with spherical lines of curvature is an open problem. Our primary objective is to close this gap and develop a construction that is efficient in controlling global properties. We will be guided by the concept of lifted-folding, which we have recently established for discrete isothermic surfaces [Hoffmann-Szewieczek (2024)]. It is a novel approach to sphericality of curvature lines, which generalizes a classical method for planar curvature lines. The underlying idea is that any such surface can be described by a holomorphic map with circular/straight folding axes and a folding function. This splitting of the data facilitates the control of global properties: periodicity and embeddedness of the spherical curvature lines are already determined by the choice of the holomorphic map. Similar to the discrete case, we expect that restrictions on the folding functions will then lead to isothermic topological cylinders and tori. Using the concept of lifted-folding we then address the question: are there embedded, non-rotational isothermic tori with spherical lines of curvature? The current state of knowledge about general embedded isothermic tori is notably limited. Contrary, a variety of remarkable works show that embedded and (up to Möbius transformations) non-rotational isothermic tori cannot be minimal or CMC in space forms. Starting from suitable holomorphic maps composed of closed elastic curves with no self-intersections, we anticipate to construct embedded isothermic tori with spherical curvature lines via lifted-folding. Our research endeavours are underpinned by preliminary work in the discrete setup: numerical examples foliated by discrete wavelike elastica suggest that embedded tori do indeed exist in this class.

• DFG Programme: Research Grants