Noncompleteness of the Weil-Petersson metric for Teichmüller space

The Teichmüller theory of harmonic maps

“…The metric is known to have negative sectional curvature, not to be complete but to be geodesically convex (see [27], [26], [28]). …”

The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces

“…It is shown that the Weil-Petersson metric is Kählerian ([1]), with negative sectional curvature ( [19], [24]). The Weil-Petersson Riemannian curvature tensor is given as the TrombaWolpert formula ( [19], [24]): [22]). This is caused by pinching off at least one short closed geodesic on the surface.…”

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

“…In [3] and [4], the authors posed the problem of developing a means by which one can study the resulting Arakelov-induced metric on M g . Specifically, it is asked if the Arakelov-induced metric on the moduli space M g is complete or not, which in the case of the classical Weil-Petersson metric was first answered in [13] and [11].…”