Surface Geometry and Integrable Systems: Theory and Experiment

• Applicant: Professor Dr. Franz Pedit
• Subject Area: Mathematics
• Term: from 2007 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 42963659

Surface geometry in 3-space provides a natural platform for the interaction and cross-fertilization of various mathematical areas: curvature relations on a surface give rise to non-linear elliptic PDEs which also are a paradigm for more general non-linear problems; Riemann surface theory, holomorphic vector bundles (Higgs bundles), and meromorphic connections play a central role in the study of special surface classes, such as constant mean curvature (model equilibrium fluid interfaces) and Willmore surfaces (elastic membranes) in space forms, and more generally in the study of harmonic maps; the integrability equations of special surfaces can often be rephrased in terms of completely integrable systems; working in low dimensions allows for computation, visualization and experimentation to test conjectures and obtain evidence of unexpected phenomena. In this context, we propose to investigate the structure and moduli spaces of special surfaces, mainly the prototypical case of constant mean curvature surfaces and harmonic maps into the 2-sphere.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry
• Participating Person: Dr. Nicholas Schmitt