Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

Abstract

We study a variational Ginzburg–Landau type model depending on a small parameter \(\varepsilon >0\) for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As \(\varepsilon \rightarrow 0\), these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus \({\mathfrak {g}}\) of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the \(2{\mathfrak {g}}\)-dimensional space of harmonic 1-forms on S if \({\mathfrak {g}}\geqq 1\). Our second main result determines the interaction energy (called renormalized energy) between vortex points as a \(\Gamma \)-limit (at the second order) as \(\varepsilon \rightarrow 0\). The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces \(S\subset {{\mathbb {R}}}^3\), in particular, to a physical model for non-tangent maps to S coming from micromagnetics.