Elizabethan vortices

TL;DR

This paper explores radial solutions to specific elliptic equations as Abelian vortices on surfaces with conical excess angles, demonstrating their global embedding in hyperbolic space through Painleve III asymptotic analysis.

Contribution

It introduces a novel interpretation of elliptic sinh-Gordon and Tzitzeica vortices as surfaces with Elizabethan ruff-like geometry and proves their hyperbolic embeddings.

Findings

Surfaces have conical excess angles at infinity.

Radial solutions correspond to Abelian vortices.

Surfaces can be embedded in hyperbolic space.

Abstract

Radial solutions to the elliptic sinh-Gordon and Tzitzeica equations can be interpreted as Abelian vortices on certain surfaces of revolution. These surfaces have a conical excess angle at infinity (in a way which makes them similar to Elizabethan ruff collars). While they can not be embedded in the Euclidean 3-space, we will show that they can be globally embedded in the hyperbolic space. The existence of these hyperbolic embeddings follows from the asymptotic analysis of a Painleve III ODE.