Non-completeness of the Arakelov-induced metric on moduli space of curves (Q2491166)

The paper investigates the relation between hyperbolic and Arakelov metrics on curves \(X\) of genus \(g\geq 2\). Recall that the holomorphic differentials on \(X\) have a natural inner product, thus producing a curvature form, Green's function, and a hermitian metric on \(X\). It differs from the hyperbolic metric by the exponential of a function on \(X\), called the conformal factor. This factor can be estimated using explicit formulas for the hyperbolic heat kernel, and the Fuchsian group. The methods come from the theory of Selberg trace formula. The metric on \(X\) induces a metric on the moduli space parametrising \(X\). As an application the author shows that for the Arakelov-induced metric on the moduli space the boundary has finite distance, so that this metric is not complete.