Canonical Bergman Metric And Quadratic Differential

Let S be a closed Riemann surface with genus g? 2, and p be the canonical Bergman metric on S. In this paper, we mainly use the canonical Bergman metric to introduce the p- inner product and the p- norm on the space of holomorphic quadratic differentials on S. We show that the unit sphere S? of the normed linear space(QQ(S),||·||?) is a compact subset. This leads to a simplified proof of the famous result that Q(S) is of finite dimension.In the first chapter, we discuss the background, objective and significance of this research, as well as the contents and methods of the research.In the second chapter, we introduce quadratic differentials, inner products of the Weil-Petersson type, the Poincare metric and the canonical Bergman metric. We discuss the similarities and differences between the Poincare metric and Bergman metric, which also motivates our research.In the third chapter, we introduce the ?- inner product and the ?- norm on Q(S). We show that the unit sphere of the normed linear space(Q(S),||·||? is a compact subset. Finally, we prove that Q(S) is a finite dimensional linear space.