Metrics with positive constant curvature and modular differential equations

TL;DR

This paper investigates differential equations involving meromorphic modular forms on the upper half-plane, linking their properties to the existence of specific conformal metrics with positive constant curvature.

Contribution

It establishes a connection between Fuchsian differential equations with modular form coefficients and the existence of conformal metrics with positive constant curvature on the upper half-plane.

Findings

Characterization of Fuchsian equations with modular form coefficients

Conditions for the existence of conformal metrics with curvature 1/2

Relation between differential equations and geometric structures

Abstract

In this paper, we consider the problem when a differential equation y"(z)=Q(z)y(z) is Fuchsian on H* and apparent on H, where Q(z) is a meromorphic modular form of weight 4 on SL(2,Z) and H denotes the complex upper half-plane. Such a problem is closely related to the problem of the existence of a conformal metric with curvature 1/2 on H.