Flat structure of meromorphic connections on Riemann surfaces

TL;DR

This paper investigates the behavior of infinite self-intersecting geodesics on Riemann surfaces with meromorphic connections, establishing relations among different mathematical structures and proving a Poincaré-Bendixson theorem under specific conditions.

Contribution

It extends the study of geodesic limit sets to infinite self-intersecting cases and proves a new Poincaré-Bendixson theorem for certain meromorphic connections.

Findings

Relation among meromorphic differentials, flat metrics, and connections established.

Poincaré-Bendixson theorem proved for infinite self-intersecting geodesics with specific monodromy.

Conditions on monodromy group G ensure the theorem's applicability.

Abstract

The possible omega limit sets of simple geodesics for meromorphic connections on compact Riemann surfaces have been studied by Abate, Tovena and Bianchi. In this paper, we study the same problem for infinite self-intersecting geodesics. In the first part of the paper we study relation among meromorphic $k$-differentials, singular flat metrics and meromorphic connections. Moreover, we prove a Poincar\'e-Bendixson theorem for infinite self-intersecting geodesics of meromorphic connections with monodromy in $G$, where $\arg G^k=\{0\}$ for some $k\in\mathbb{N}$.