Geodesic incompleteness in the CP^1 model on a compact Riemann surface

TL;DR

This paper proves that the moduli space of static solutions in the CP^1 model on a compact Riemann surface is geodesically incomplete, implying finite-time collapse of lumps into singularities.

Contribution

It establishes the geodesic incompleteness of the moduli space for the CP^1 model on any compact Riemann surface, highlighting potential singularity formation.

Findings

Moduli space is geodesically incomplete

Lumps can collapse in finite time

Singularities can form in the model

Abstract

It is proved that the moduli space of static solutions of the CP^1 model on spacetime Sigma x R, where Sigma is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.