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的环面。如果表面积在保持封闭体积的无穷小扰动下不改变，则表面具有恒定的平均曲率。环面是看起来大致（即大）像甜甜圈表面的表面。像Heinz Hopf这样的经典微分几何家研究了平均曲率恒定的环面是否存在，直到1986年Henry Wente最终通过构造这样的曲面解决了这个问题。基于非线性偏微分方程的解和代数几何数据之间的微妙对应关系，文特的构造导致了几何分析的新方法。这种对应关系只适用于某些偏微分方程，这些偏微分方程可以与所谓的可积系统联系起来。该项目的主要目标是证实乌尔里希·平卡尔的猜想。这个猜想表明，在任何封闭曲线的任意小管状邻域中，存在如此多的平均曲率恒定的环面。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。而相应的非线性偏微分方程属于另一个可积系统。这个观察结果引起了这样一个问题：微分方程的解和代数数据之间的微妙对应是否可能通过适当的极限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，人们对各种可积系统的微妙对应已经有了很好的理解。特别是，新的方法已经发展到变形的代数数据，其中对应的解的可积系统被保留。我们要对这些方法进行推广，通过取适当的极限，将一个可积系统的解的代数数据转化为另一个可积系统的解的代数数据。本课题研究的是等平均曲率曲面的代数数据的变形。对于闭曲线系统，代数数据和相应的解都有一定的极限，从而产生了微妙对应的代数数据和两边的解。

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.