Simple Closed Curves And Their Applications In Riemannn Surface

The problem of how to parametrize the variations of complex structures on a fixed base surface has been studied by Riemann. This problem is called Riemann’s moduli problem. It is difficult to grasp the substance of Riemann’s moduli problem. Fortunately, by considering the universal covering space of the moduli space, Teichmuller succeeded in this problem. This universal covering space is called the Teichmuller space by Ahlfors.During the last few decades, Teichmuller theory has gained great success, and become playing more and more important roles in many areas in Mathematics. Many Mathematicians made contributions to the theory. Among those, by considering the geometric intersection number of simple closed curves(and more generally measured foliations) on the surface,Thurston has given a compactification of the Teichmuller space, which is now known as the Thurston compactification.In this paper, with the aid of the algebraic intersection number of simple closed curves,we will study properties of simple closed curves and of Riemann surfaces.The paper is organized as follows, with three chapters.The first chapter which is the introduction presents backgrounds and current status of researches. A brief introduction to our work is also given.The second chapter gives relevant preknowledges of Thurston’s compactification of the Teichmuller space(where the notions such as Frike space, pants decomposition and holomorphic quadratic differentials are involved in).In the third chapter, we will study properties of simple closed curves. There are two natural ways to count the number of intersection points between two simple closed curves,which are the algebraic intersection number and the geometric intersection number. For any two simple closed curves ? and ?, the geometric intersection number is, by definition, the infimum of the number of intersection points of1 L and2L, wherejL is taken over the freehomotopy class of ? and ? respectively.In this paper, we will use the algebraic intersection number between two homology classes of simple closed curves. Generally speaking, direct computation of such an intersection number is difficult. In this paper, we will construct a mapping which sends simple closed curves into some projective space, where it becomes easy to deal with the images of simple closed curves.