Canonical Metrics on the Moduli Space of Riemann Surfaces I

One of the main purpose of this paper is to compare those well-known canonical and complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new metrics, the Ricci metric and the perturbed Ricci metric. We will prove that these metrics are equivalent to those classical complete metrics. For this purpose we study in detail the asymptotic behaviors and the signs of the curvatures of these new metrics. In particular we prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic sectional curvature and bounded bisectional and Ricci curvature. The study of the Teichmüller spaces and moduli spaces of Riemann surfaces has a long history. It has been intensively studied by many mathematicians in complex analysis, differential geometry, topology and algebraic geometry for the past 60 years. They have also appeared in theoretical physics such as string theory. The moduli space can be viewed as the quotient of the corresponding Teichmüller space by the modular group. There are several classical metrics on these spaces: the Weil-Petersson metric, the Teichmüller metric, the Kobayashi metric, the Bergman metric, the Caratheodory metric and the Kähler-Einstein metric. These metrics have been studied over the years and have found many important applications in various areas of mathematics. Each of these metrics has its own advantages and disadvantages in studying different problems. The Weil-Petersson metric is a Kähler metric as first proved by Ahlfors, both of its holomorphic sectional curvature and Ricci curvature have negative upper bounds as conjectured by Royden and proved by Wolpert. These properties have found many applications by Wolpert, and they were also used in solving problems from algebraic geometry by combining with the Schwarz lemma of Yau ([5], [17]). But as first proved by Masur it is not a complete metric which prevents the understanding of some aspects of the geometry of the moduli spaces. Siu