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Weil–Petersson Teichmüller space III: dependence of Riemann mappings for Weil–Petersson curves

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Abstract

The classical Riemann mapping theorem implies that there exists a so-called Riemann mapping which takes the upper half plane onto the left domain bounded by a Jordan curve in the extended complex plane. The primary purpose of the paper is to study the basic problem: how does a Riemann mapping depend on the corresponding Jordan curve? We are mainly concerned with those Jordan curves in the Weil–Petersson class, namely, the corresponding Riemann mappings can be quasiconformally extended to the whole plane with Beltrami coefficients being square integrable under the Poincaré metric. After giving a geometric characterization of a Weil–Petersson curve, we endow the space of all normalized Weil–Petersson curves with a new real Hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex Hilbert manifold structure, which implies that an appropriately chosen Riemann mapping depends continuously on a Weil–Petersson curve (and vice versa). This can be considered as the first result about the continuous dependence of Riemann mappings on non-smooth Jordan curves.

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Notes

  1. We conjecture that in general neither \(\log f'\) nor \(\log (h^{-1})'\) depends continuously on \(\Gamma \) (or b).

  2. After an earlier version of this manuscript was posted on arXiv [46], we learned from Tim Mesikepp that Bishop [6] obtained various geometric characterizations of a bounded Weil-Petersson curve. The authors would like to thank Tim Mesikepp for calling this reference to their attention.

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Acknowledgements

The authors would like to thank the referee for a careful reading of the manuscript and for several corrections which improves the presentation of the paper.

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  1. Department of Mathematics, Soochow University, Suzhou, 215006, People’s Republic of China

    Yuliang Shen & Li Wu

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  1. Yuliang Shen
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Correspondence to Yuliang Shen.

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Communicated by Ngaiming Mok.

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Shen, Y., Wu, L. Weil–Petersson Teichmüller space III: dependence of Riemann mappings for Weil–Petersson curves. Math. Ann. 381, 875–904 (2021). https://doi.org/10.1007/s00208-020-02067-5

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