The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps

TL;DR

This paper extends the geodesic approximation method to harmonic maps, proving a coercivity property of the Hessian in sigma models, and explores the global coercivity and dynamics of CP^1 lumps on S^2.

Contribution

It establishes a coercivity property of the Hessian for general sigma models and analyzes the dynamics of CP^1 lumps, including numerical evidence for coercivity in certain sectors.

Findings

Proves coercivity of the Hessian for sigma models with Kähler domains and targets.

Shows the Hessian fails to be globally coercive in the degree 1 sector of CP^1 on S^2.

Numerical evidence suggests global coercivity in higher degree sectors for n>1.

Abstract

The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximation, and hence proved that it is valid in the low speed regime. His method employs energy estimates which rely on a key coercivity property of the Hessian of the energy functional of the theory under consideration. In this paper we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact K\"ahler domain and target. We go on to prove a continuity property for our result, and show that, for the CP^1 model on S^2, the Hessian fails to be globally coercive in the degree 1 sector. We present numerical evidence which suggests that the Hessian is globally coercive in a certain equivariance class of the degree n sector for n>1. We also prove that, within the geodesic approximation, a single CP^1 lump moving on S^2 does not generically travel on a great circle.