Canonical Metrics on the Moduli Space of Riemann Surfaces II

By Kefeng Liu, Xiaofeng Sun, Shing-Tung Yau. 2004

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One of the main purpose of this paper is to compare those well-known canonical and complete metrics on the Teichmuller 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 Kahler metric with bounded negative holomorphic sectional curvature and bounded bisectional and Ricci curvature. The study of the Teichmuller 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 Teichmuller space by the modular group. There are several classical metrics on these spaces: the Weil-Petersson metric, the Teichmuller metric, the Kobayashi metric, the Bergman metric, the Caratheodory metric and the Kahler-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 Kahler 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

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b2fc063cab4d48e26d76f1594600fbdaad6bab61

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Citations in the corpus

Geometry of domains with the uniform squeezing property (2009, background): “[Ga]). After the appearance of the paper, Kefeng Liu and Shing-Tung Yau have kindly informed the author that they have named a similar domain a ‘holomorphic homogeneous regular’ one, without emphasizing the dependence on a and b, in [ LSY ]”
A general Schwarz Lemma for almost-Hermitian manifolds (2007, background): “This proved to be extremely useful in differential geometry and complex analysis (see, for example, [17]).”
An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (2013, background): “One might also see the following works: [11], [39], [48], [50], [35], [36], and [56].”
The geometry of right angled Artin subgroups of mapping class groups (2010, unknown): “However, each of these is quasi-isometric to the Teichmüller metric (see [36, 42, 29, 30]), and so the conclusion of Theorem 1.”
PROPERTIES OF SQUEEZING FUNCTIONS AND GLOBAL TRANSFORMATIONS OF BOUNDED DOMAINS (2013, background): “, Teichmu¨ller spaces of compact Riemann surfaces are homogenous regular. In the past decade, comparisons of various intrinsic metrics on Teichmu¨ller spaces were extensively studied (see e.g. [4][13][14][23]). The equivalence of certain intrinsic measures on Teichmu¨ller spaces was proved in [18]. ”
On the uniform squeezing property of bounded convex domains in ℂn (2016, background): “The notion of holomorphic homogeneous regular, or equivalently, uniformly squeezing for complex manifolds has been introduced in [Liu et al. 2004; 2005] and [Yeung 2009], respectively.”