Geodesic completeness for meromorphic metrics: the case of coercive ones

Abstract

This thesis is concerned with extending the idea of geodesic completeness from pseudo-Riemannian to complex geometry: we take, however a completely holomorphicpoint of view; that is to say, a 'metric' will be a (meromorphic) symmetric section of the twice covariant holomorphic tensor bundle. We shall hint at the need of reformulating some aspects of the theory of differential equations in the complex domain, originating in the interpenetration betwixt differential and algebraic aspects when analytical continuation is pursued. The notion itself of path should be reformulated: we remark that geodesics will be defined on Riemann surfaces which are domains over regions in the complex plane. Of course geodesics will be eventually defined to be auto-parallel paths, but we shall focus our attention on the fact that the Levi-Civita connection will be meromorphic if the metric from which it is induced is allowed to have poles or even simply to lower somewhere in its rank. We shall study rather more deeply a class of manifolds, namely warped products of Riemann surfaces; some hypotheses concerning their metrics will be done, (metrics will be suppose to be 'coercive', in a sense that will be defined) but we shall show that the range of applicability of the yielded completeness theorems will not be exceedingly restricted.