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Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

  • Martin Bridgeman and Yunhui Wu
Published/Copyright: April 16, 2020

Abstract

In this article we show that for every finite area hyperbolic surface X of type ( g , n ) and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to - g as the genus g goes to infinity.

Award Identifier / Grant number: DMS-1564410

Funding statement: Martin Bridgeman’s research was supported by NSF grant DMS-1564410. Yunhui Wu is partially supported by China’s Recruitment Program of Global Experts.

Acknowledgements

The authors would like to thank Jeffrey Brock, Ken Bromberg and Michael Wolf for helpful conversations on this project.

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Received: 2019-08-18
Revised: 2020-02-12
Published Online: 2020-04-16
Published in Print: 2021-01-01

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