Constrained Willmore surfaces

浸没的Willmore泛函或弹性弯曲能W -∫H2在当代表面理论和从膜的生物物理学到弦理论的应用中是一个重要的全局不变量。我们研究了所谓的约束Willmore曲面，它是Riemann曲面的保形浸入，是紧支撑无穷小保形变化下VV的临界点。在项目的第三阶段，我们的研究重点是寻找约束Willmore tori的Willmore能量的估计。这些都可以用复杂代数几何的方法来构造，这表明基本的代数几何不变量，如某些附属全纯映射的程度或所谓的谱曲线的属，应该与浸入的威莫能量有关。对于一些特殊情况，我们可以建立这样的联系，我们的计划是将得到的估计推广到任意约束的Willmore环面。

The Willmore functional or elastic bending energy W — ∫ H2 of an immersion is a global invariant of fundamental importance in contemporary surface theory and applications ranging from the biophysics of membranes to string theory. We investigate the so-called constrained Willmore surfaces which are conformal immersions of a Riemann surface that are critical points of VV under compactly supported infinitesimal conformal variations. During the third period of the project the focus of our investigations is on finding estimates for the Willmore energy of constrained Willmore tori. These can all be constructed by methods of complex algebraic geometry which suggests that the fundamental algebraic geometric invariants like the degree of certain appendant holomorphic maps or the genus of the so called spectral curve should be related to the Willmore energy of the immersion. For several special cases we can establish such a link and our plan is to generalize the obtained estimates to arbitrary constrained Willmore tori.