The soul curve of cmc tori

本研究项目涉及的最重要的对象是常平均曲率环面。如果一个曲面的总面积在保持封闭体积不变的无穷小扰动下不变，则该曲面的平均曲率不变。圆环面是看起来大致(即大体上)像甜甜圈表面的表面。像Heinz Hopf这样的经典微分几何学家研究了常平均曲率环面是否确实存在，直到Henry Wente在1986年通过构造这样的曲面最终解决了这个问题。Wente的构造导致了几何分析的新方法，该方法基于非线性偏微分方程解和代数几何数据之间的微妙对应。这种对应关系只适用于某些偏微分方程组，这些偏微分方程组可以与所谓的可积系统联系在一起。该项目的主要目的是证实Ulrich Pinkall提出的一个猜想。这一猜想表明，有如此多的常平均曲率环面，以至于在任何闭合曲线的任意小管状邻域中都存在一个。我们的方法还利用了另一种形式的微妙对应，即闭合曲线和代数几何数据之间的对应。然而，相应的非线性偏微分方程属于另一个可积系统。这种观察引出了这样一个问题：通过采取适当的限制，微分方程解和代数数据之间的微妙对应是否可能从一个可积系统传递到另一个可积系统。在过去的三十年里，对于各种可积系统，人们已经很好地理解了微妙对应。特别是，已经发展了新的方法来变形代数数据，其中保留了相应解的可积系。我们想要扩展这些方法，以便通过取适当的极限，它们将一个可积系统的解的代数数据变形为另一个不同可积系统的解的代数数据。该研究项目研究常平均曲率曲面的代数数据的变形。代数数据和相应解的某些限制导致了闭合曲线系的微妙对应两侧的代数数据和解。

The most important objects involved in this research project are tori of constant mean curvature. A surface has constant mean curvature if the total area does not change under infinitesimal perturbations which preserve the enclosed volume. Tori are surfaces which look roughly (i.e. in the large) like the surface of a donut. Classical differential geometers like Heinz Hopf investigated whether tori of constant mean curvature in fact exist, until Henry Wente finally settled this question in 1986 by constructing such surfaces. Wente's construction led to new methods in geometric analysis, based on a subtle correspondence between solutions of nonlinear partial differential equations and algebraic geometric data. This correspondence is applicable only to certain partial differential equations, which can be associated with so-called integrable systems.The main objective of the project is to confirm a conjecture due to Ulrich Pinkall. This conjecture states that there are so many tori of constant mean curvature that there exists one in any arbitrarily small tubular neighbourhood of any closed curve. Our approach also utilises another form of the subtle correspondence, namely between closed curves and algebraic geometric data. The corresponding nonlinear partial differential equation belongs to another integrable system, however. This observation gives rise to the question whether it is possible for the subtle correspondence between solutions of differential equations and algebraic data to pass from one integrable system to another by taking suitable limits. Within the last thirty years the subtle correspondence has been understood quite well for various integrable systems. In particular, new methods have been developed to deform the algebraic data, where the integrable system of the corresponding solutions is preserved. We want to extend these methods, such that by taking a suitable limit, they deform the algebraic data of solutions of one integrable system into algebraic data of solutions of a different integrable system. The research project investigates deformations of algebraic data of surfaces of constant mean curvature. Certain limits of both the algebraic data and the corresponding solutions result in the algebraic data and the solutions on both sides of the subtle correspondence for the system of closed curves.