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

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 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 Open image in new window of compact Riemann surfaces of genus g. We prove here that Open image in new window