Constant mean curvature surfaces of higher genus

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The study of surfaces in 3-dimensional space forms with special geometric properties is an important part of differential geometry. Among the most interesting are those surfaces critical with respect to the area functional (under fixed enclosed volume), the minimal surfaces respectively the constant mean curvature (CMC) surfaces. CMC surfaces are determined by their associated family of flat special linear connections on a fixed rank 2 bundle. In the case of CMC tori these flat connections can be parametrized in terms of the spectral data because the generic flat connection on a torus reduces to the sum of flat line bundle connections due to the abelian nature of the first fundamental group of a torus. As a consequence, CMC tori are parametrized in terms of algebro-geometric data. By these integrable system methods, many new examples of compact CMC and minimal tori have been constructed. More recently also questions concerning the moduli space of solutions have begun to be investigated in detail.On the other hand, there are only few known examples of compact CMC and minimal surfaces of higher genus. Notably, the known examples like the Lawson surfaces are only given implicitly as solutions to non-linear partial differential equations. In recent work, we have extended the integrable systems approach to symmetric CMC surfaces of higher genus and developed a spectral curve representation for those surfaces. This yields new tools for a systematic study of CMC and minimal surfaces of higher genus and their moduli spaces.The goal of the project is to improve our understanding of CMC and minimal surfaces of higher genus by integrable systems methods. Of particular importance is the enhancements of the construction of admissible spectral data. For this purpose, we will study the generalized Whitham flow in detail. Its long time existence would yield new families of CMC surfaces of higher genus and would lead to a rigorous description of the moduli space of embedded symmetric CMC surfaces of genus 2.Another goal is the extension of the spectral curve approach to general compact surfaces via abelianization of the moduli space of flat connections, and to develop a method, which generates spectral data of non-symmetric CMC surfaces of higher genus.As a byproduct of our explicit approach to CMC surfaces, we have been able to carry through numerical experiments with higher genus CMC surfaces. These have led to a detailed picture of the moduli space of CMC surfaces of genus 2 in the 3-sphere with symmetries. A natural task is to extend our experiments to higher genus surfaces without symmetries in order to obtain reliable conjectures concerning the moduli space of compact CMC surfaces.