Projective structures on curves and conformal geometry

TL;DR

This paper explores the theory of projective structures on curves, clarifies their definitions and classifications, and applies this understanding to demonstrate the non-existence of solutions to the Yamabe problem in certain conformal settings.

Contribution

It provides a detailed correspondence between various definitions of projective structures and corrects inaccuracies in existing literature.

Findings

Clarifies the equivalence of different definitions of projective structures.

Corrects inaccuracies in the literature regarding projective structures.

Shows the Yamabe problem for curves generally has no solutions.

Abstract

Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we describe in detail the correspondence between different equivalent definitions of projective structures and their isomorphism classes, correcting long-standing inexactitudes in the literature. As an application, we show that the {\em Yamabe problem for curves} in a conformal/M\"obius ambient space has no solutions in general.