Combinatorial structures on Riemann surfaces and topological properties of the moduli space.

点击翻译按钮获取中文摘要

Our results are as follows :1.We consider Riemann surfaces with Abelian differential constructed from lightning pairs. A lightning pair is a pieacewise linear loop in the complex plane determined by a certain kind of combinatorial data. We give a method of obtaining from the combinatorial data of a lightning pair the genus of the resulting Riemann surface.2.We give a sufficient condition for the completion of the form which is induced from a pre-Tango structure to have non-closed global differential 1-forms. Moreover, we give a lower bound for the dimension of the locus of the curves which have pre-Tango structures inducing such completions, in the moduli space of curves.3.A Haefliger (6,3)-knot means a smoothly embedded 3-sphere in the 6-sphere. We give a definition of unknotting numbers of Haefliger (6,3)-knots geometrically, and determine the unknotting number of each Haefliger (6,3)-knot.4.Twisting the Killing vector fields of certain kind of Kerr-Ads black holes, we reproduce the compact Sasaki-Einstein manifolds constructed by Gauntlett, Martelli, Sparks and Waldram. We also discuss an implication of the twist in string theory and M-theory.5.We construct explicitly a new infinite series of Einstein metrics on the S^3-bundles over S^2, which containing infinite numbers of inhomogeneous ones.