Open Access
Jan 2005 Canonical metrics on the moduli space of Riemann Surfaces II
Kefeng Liu, Xiaofeng Sun, Shing–Tung Yau
J. Differential Geom. 69(1): 163-216 (Jan 2005). DOI: 10.4310/jdg/1121540343


In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the Kähler–Einstein metric, we show that the logarithmic cotangent bundle over the Deligne–Mumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the Kähler–Einstein metric by using the Kähler–Ricci flow and a priori estimates of the complex Monge-Ampere equation.


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Kefeng Liu. Xiaofeng Sun. Shing–Tung Yau. "Canonical metrics on the moduli space of Riemann Surfaces II." J. Differential Geom. 69 (1) 163 - 216, Jan 2005.


Published: Jan 2005
First available in Project Euclid: 16 July 2005

zbMATH: 1086.32011
MathSciNet: MR2144543
Digital Object Identifier: 10.4310/jdg/1121540343

Rights: Copyright © 2005 Lehigh University

Vol.69 • No. 1 • Jan 2005
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