| Morse theory is a useful tool in differential topology. Recently, Morse theory becomes a hot spot of mathematical research. Meanwhile, Morse theory solved a series of problems that is difficult in other mathematical subjects. Morse theory provides lots of analytical methods and the overall decomposition approach, with which people can gain deeper understanding of a differential manifold with its geometrical and topological properties. Recently, Morse theory's applications in the theory of partial differential equations and Lie groups gained more people's attention.But unfortunately, Morse theory and another very useful subject in mathematical: single complex analysis theory had not significant connection. This paper attempts to use the methods that construct a Morse function on one-dimensional complex manifold and by noticing the influence in the whole structure of the manifold caused by Morse theory, We reproved a very important theorem on Riemann surface: Uniformization theorem,which is very important in the field of single-variable complex analysis. With the help of universal cover, we may be able to give a good model for differential structure of Riemann surface and Teichmuller space which is of elegant structure. It shows the obvious influence on Riemann surface's complex structure caused by Morse function.In addition, this paper also put forward a generalized Morse lemma and proved it. We promoted the Morse lemma which originally just effective in the situation of single-variable to the situation that is effective in several variables. We put forward a sufficient condition which is relatively easy to validate, and we gave its proof. Then Morse lemma has been promoted dramatically.The basic work of this paper includes the following aspects:(1)We introduced Morse theory by detail.(2)We reproved Unifromization theorem by Morse theory.(3)We popularized Morse lemma and put forward a sufficient condition for its promotion.(4) We put forward a right-equivalent descriptions on the edge of an object in the image by using Morse lemma so that we can make it normalized. (5)We demonstrated our algorithm by an example. |
