Non-Completeness of Zograf–Takhtajan's Kähler Metric for Teichmüller Space of Punctured Riemann Surfaces

Abstract:

Zograf and Takhtajan introduced a new Kähler metric on the Teichmüller space T _g , _n (n>0), in calculating the first Chern form of the Quillen metric for families of -operators. The metric is described in terms of the Eisenstein–Maass series. We prove that it is incomplete. And we also give an alternative proof of non-completeness of the Weil–Petersson metric. For that, we use the pinching family, constructed by Wolpert, whose tangent vectors are always represented by using the relative Poincaré series associated with the pinched geodesic.