The soul curve of cmc tori

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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.