Jacobian Estimate

Abstract

In this chapter we show that the vortex balls provided by Theorem 4.1, although they are constructed through a complicated process and are not completely intrinsic to (u,A) (and not unique), have in the end a simple relation to the configuration (u,A), namely that the measure Σ_i2πd_iδ_ai is close in a certain norm to the gauge-invariant version of the Jacobian determinant of u, an intrinsic quantity depending on (u,A). This will allow us, in the next chapters, to extract from G_ε(u,A), in addition to the vortex energy πΣ_i |d_i||log ε| contained in the vortex balls, a term describing vortex-vortex interactions and vortex-applied field interactions in terms of the measure Σ_i2πd_iδ_ai.