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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Equiboundedness of the Weil-Petersson metric

Author: Scott A. Wolpert
Journal: Trans. Amer. Math. Soc. 369 (2017), 5871-5887
MSC (2010): Primary 32G15, 30F60
Published electronically: April 24, 2017
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Abstract: Given a topological type for surfaces of negative Euler characteristic, uniform bounds are developed for derivatives of solutions of the $ 2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the $ C^0$, $ C^{2,\alpha }$ and $ L^2$ norms for harmonic Beltrami differentials are analyzed. Uniform bounds are given for the covariant derivatives of the Weil-Petersson curvature tensor in terms of the systoles of the underlying Riemann surfaces and the projections of the differentiation directions onto pinching directions. The main analysis combines Schauder and potential theory estimates with the analytic implicit function theorem.

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Scott A. Wolpert
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Received by editor(s): November 28, 2015
Received by editor(s) in revised form: May 26, 2016
Published electronically: April 24, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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