Geodesic completeness for some meromorphic metrics

Geodesic completeness for meromorphic metrics: the case of coercive ones

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 geo…

具有实周期的Fuchsian亚纯联络的动力学

Dynamics of Fuchsian meromorphic connections with real periods

该研究聚焦具有实周期的Fuchsian亚纯联络的测地线动力学，借助奇异平坦度量、极点附近测地线刻画及推广的Teichmüller引理，精确给出了简单测地线的ω-极限集特征。

A Holomorphic Point of View about Geodesic Completeness

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that our definition sheds a bit of new light on the behaviour of 'singularities' of geodesics in space-time. We also show that some geodesics, which 'end' at finite time in the classical sense, can be naturally continued besides their ends. As a matter of fact, complex metrics naturally show a meromo…

A Poincaré-Bendixson theorem for meromorphic connections on Riemann surfaces

We prove a Poincaré-Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.

黎曼曲面上亚纯联络的平坦结构

Flat structure of meromorphic connections on Riemann surfaces

本文研究紧黎曼曲面上亚纯联络的无穷自相交测地线的 omega 极限集，建立其与亚纯 k-微分和奇异平坦度量的关系，并证明在特定单值群条件下的一类 Poincaré-Bendixson 定理。

Clifton-Pohl torus and geodesic completeness by a 'complex' point of view

We show that a natural complexification and a mild generalization of the idea of completeness guarantee geodesic completeness of Clifton-Pohl torus; we explicitely compute all of its geodesics.