Geodesic completeness for some meromorphic metrics
  In this paper we investigate possible extensions of the idea of geodesic completeness in complex manifolds, following two directions: metrics are somewhere allowed not to be of maximum rank, or to have 'poles' somewhere else. Geodesics are eventually defined on Riemann surfaces over regions in the Riemann sphere. Completeness theorems are given in the framework of warped products of Riemann surfaces. 
Claudio Meneghini