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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 μhyp,X, arising from hyperbolic geometry, and the Arakelov metric form μAr,X, arising from arithmetic algebraic geometry. Now consider a sequence Xt 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.

Non-completeness of the Arakelov-induced metric on moduli space of curves