Constant mean curvature surfaces of higher genus

研究三维空间形式中具有特殊几何性质的曲面是微分几何的重要组成部分。其中最有趣的是那些表面的关键方面的功能（在固定的封闭体积），最小的表面分别恒定平均曲率（CMC）的表面。CMC曲面是由它们在固定秩2丛上的相关平坦特殊线性联络族确定的。在CMC环面的情况下，这些平坦连接可以根据谱数据参数化，因为环面上的一般平坦连接由于环面的第一基本群的阿贝尔性质而减少为平坦线束连接的总和。因此，CMC环面参数化的代数几何数据。通过这些可积系统方法，构造了许多新的紧CMC和最小环面的例子。最近也开始详细研究有关的模空间的解决方案的问题。另一方面，只有少数已知的例子，紧凑CMC和极小曲面的高亏格。值得注意的是，已知的例子，如劳森表面只隐含地给出了非线性偏微分方程的解决方案。在最近的工作中，我们已经扩展了可积系统的方法，对称CMC曲面的高亏格，并开发了这些曲面的谱曲线表示。这为系统地研究CMC和高亏格极小曲面及其模空间提供了新的工具，本项目的目标是通过可积系统的方法来提高我们对CMC和高亏格极小曲面的理解。特别重要的是，可接受的光谱数据的建设的增强。为此，我们将详细研究广义Whitham流。它的长期存在将产生新的更高亏格的CMC曲面族，并将导致亏格2的嵌入对称CMC曲面的模空间的严格描述。另一个目标是通过平坦联络的模空间的阿贝尔化将谱曲线方法推广到一般紧曲面，并发展了一种方法，该方法生成了高亏格的非对称CMC曲面的谱数据，作为CMC曲面显式方法的副产品，我们已经能够对高亏格的CMC曲面进行数值实验。这些导致了一个详细的图片的CMC表面的模空间的亏格2在3球对称。一个自然的任务是将我们的实验扩展到没有对称性的更高亏格的曲面，以获得关于紧凑CMC曲面的模空间的可靠描述。

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.