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

It is proved that the moduli space of static solutions of the C P 1 model on spacetime Σ × R , where Σ 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. Let M n denote the moduli space of degree n static solutions of the C P 1 model on a compact Riemann surface Σ, that is, on spacetime Σ × R . The kinetic energy functional induces a natural metric g on M n , and geodesics of the Riemannian manifold ( M n , g ), when traversed at slow speed, are thought to be close to low energy dynamical solutions of the model, in which n lumps move slowly and interact on Σ (this is the geodesic approximation of Manton [1]). It has recently been proved that M 1 and M 2 , in the cases Σ = S 2 and Σ = T 2 respectively, are geodesically incomplete [2, 3], meaning that, according to this approximation, lumps on these surfaces can collapse to form singularities in finite time. The purpose of this note is to prove the following generalization of these results: