Blow up analysis for a cosmic strings equation

TL;DR

This paper analyzes the blow up behavior of solutions to a semilinear elliptic equation modeling cosmic strings, revealing new phenomena and connections to geometric uniformization problems.

Contribution

It develops a novel blow up analysis for solutions of a cosmic strings equation, linking blow up profiles to geometric problems involving Riemann sphere uniformization.

Findings

Blow up profiles can be characterized by differential problems with geometric significance.

New phenomena in blow up behavior are identified for solutions involving exponential nonlinearities.

Connections to the uniformization of Riemann spheres with conical singularities are established.

Abstract

In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new and surprising phenomena. For example in some cases, the blow up profile of the solution is described in terms of a differential problem that bares a geometrical meaning in the context of the uniformization of the Riemann sphere with conical singularities.