紧黎曼面上的一形式与奇性可约椭圆度量

在紧黎曼面上给定有限个点与相应的角度， 是否存在于给定点具有相应锥角度的奇性椭圆度量？这是一个有着一百多年历史，但仍未解决的经典问题。从代数的角度出发，申请人与合作者将奇性椭圆度量分成可约和不可约两类, 并且发现可约椭圆度量问题可以约化为(仅有单极点的亚纯)一形式的研究。另一方面，三十多年前人们发现紧黎曼面上的全纯一形式的模空间与动力系统中的区间交换映射以及有理多边形上的桌球运动之间的联系以来，数学家们在这个领域已建立比较完整的理论。我们计划探究紧黎曼面的复结构与该黎曼面上的一形式的极点及其留数，零点及其重数之间的非平凡关系，研究满足适当条件的一形式的模空间的性质。通过完成这项研究，我们不但可以得到可约椭圆度量存在的充分必要条件与清晰理解其模空间，还可以尝试将全纯一形式的理论推广到一形式。

Given finitely many points and the corresponding angles on a compact Riemann surface, does there exist a singular elliptic metric on the surface with the corresponding conical angles at the given points? It is a classical and open problem with a history of more than 100 years..From the algebraic viewpoint, the applicant and his collaborators divide singular elliptic metrics into two classes --- reducible metrics and irreducible ones. Moreover, we reduce the problem about reducible metrics into that of (meromorphic) 1-forms (with at most simple poles). On the other hand, mathematicians have built a relatively complete theory on moduli of holomorphic 1-forms on compact Riemann surfaces since their connections with interval exchange transformations and billiards on rational polygons in Dynamical Systems were discovered more than 30 years ago. The applicant plans to investigate the non-trivial relationship among complex structures of compact Riemann surfaces, the poles and their residues and the zeroes of their multiplicities of 1-forms living on the Riemann surfaces and to study the moduli of 1-forms satisfying suitable conditions. If the plan were completed, we would not only obtain the necessary and sufficient condition for the existence of reducible elliptic metrics and make a clear understanding of moduli of reducible metrics, but also try to generalize partially the theory of holomorphic 1-forms to 1-forms.