Constrained Willmore surfaces

我们要研究三维空间中的一类曲面，即所谓的“受限Willmore曲面”。这些被定义为Willmore函数的临界点，当只有这些变化被允许保持表面的保形类型时。这类曲面在环境空间Möbius变换下是不变的。例子包括欧几里得空间、双曲空间和3球中的常平均曲率曲面。我们的主要兴趣是新的紧实例的构造和分类，主要是拓扑球体和环面。我们希望从孤子理论的方法中得到所有约束Willmore球和环面的显式公式。

We want to study a class of surfaces in 3-space, the so-called "constrained Willmore surfaces". These are defined as the critical points of the Willmore function when only those variations are allowed that preserve the conformal type of the surface. This class of surfaces is invariant under Möbius transformations of the ambient space. Examples include constant mean curvature surfaces in Euclidean space, hyperbolic space and the 3-sphere. Our main interest is the construction and classification of new compact examples, mainly topological spheres and tori. We hope to obtain explicit formulas for all constrained Willmore spheres and tori from a soliton theoretic approach.