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Comparison Moduli Spaces of Riemann Surfaces

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Complex Analysis and Dynamical Systems

Part of the book series: Trends in Mathematics ((TM))

Abstract

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmüller theory, and illustrate how a wide range of phenomena in complex analysis are captured by this notion of moduli space. The paper includes a list of open problems in classical and modern function theory and Teichmüller theory ranging from general theoretical questions to specific technical problems.

Dedicated to the memory of Alexander Vasil’ev

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Notes

  1. 1.

    After this chapter was submitted in May 2016, the paper [80] of Yanagishita appeared, in which it was shown that for surfaces satisfying Lehner’s condition, the Weil-Petersson metric is indeed Kähler and the sectional and Ricci curvatures are negative. The convergent Weil-Petersson metric was obtained independently of Radnell, Schippers and Staubach [47, 51]. Yanagishita [80] also showed that the complex structure from harmonic Beltrami differentials is compatible with the complex structure from the Bers embedding. When combined with the results of [51], this apparently shows that these two complex structures are equivalent to that obtained from fibrations over the compact surfaces for surfaces of type (g, n).

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Acknowledgements

The authors are grateful to David Radnell for a fruitful collaboration and valuable discussions through the years, and for his comments and suggestions concerning the initial draft of the manuscript.

Eric Schippers and Wolfgang Staubach are grateful for the financial support from the Wenner-Gren Foundations. Eric Schippers is also partially supported by the National Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada, R3T 2N2

    Eric Schippers

  2. Department of Mathematics, Uppsala University, Box 480, Uppsala, 751 06, Sweden

    Wolfgang Staubach

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  1. Eric Schippers
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  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Mark Agranovsky

  2. Department of Mathematics, Holon Institute of Technology, Holon, Israel

    Anatoly Golberg

  3. Department of Mathematics, ORT Braude College, Karmiel, Israel

    Fiana Jacobzon

  4. Department of Mathematics, Holon Institute of Technology, Holon, Israel

    David Shoikhet

  5. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Lawrence Zalcman

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Schippers, E., Staubach, W. (2018). Comparison Moduli Spaces of Riemann Surfaces. In: Agranovsky, M., Golberg, A., Jacobzon, F., Shoikhet, D., Zalcman, L. (eds) Complex Analysis and Dynamical Systems. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70154-7_13

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