Abstract
Let X be a compact Riemann surface of genus g>1. We study two different, naturally defined metric forms on X: The hyperbolic metric form μ hyp, X , arising from hyperbolic geometry, and the Arakelov metric form μ Ar, X , arising from arithmetic algebraic geometry. Now consider a sequence X t of Riemann surfaces approaching the Deligne-Mumford boundary of the moduli space of compact Riemann surfaces of genus g. We prove here that
As a corollary of this result, we prove that the Weil-Petersson metric on the moduli space induced from the Arakelov metric is not complete, i.e., certain boundary components of the Deligne-Mumford compactification are at finite distance.
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The first author acknowledges support from grants from the NSF and PSC-CUNY. The second author thanks the Centre de Recerca Matemàtica (CRM) in Barcelona for its support and hospitality.
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Jorgenson, J., Kramer, J. Non-completeness of the Arakelov-induced metric on moduli space of curves. manuscripta math. 119, 453–463 (2006). https://doi.org/10.1007/s00229-006-0625-2
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DOI: https://doi.org/10.1007/s00229-006-0625-2