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

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 article we prove an analogous coercivity property for the Hessian of the energy functional of a general sigma model with compact Kähler domain and target. We go on to prove a continuity property for our result, and show that, for the CP¹ model on S², 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≥2. We also prove that, within the geodesic approximation, a single CP¹ lump moving on S² does not generically travel on a great circle. © 2003 American Institute of Physics.

Duke Scholars

Author Mark Haskins Mathematics