Renormalized energy between fractional vortices with topologically induced free discontinuities on 2-dimensional Riemannian manifolds

Abstract

On a two-dimensional Riemannian manifold without boundary we consider the variational limit of a family of functionals given by the sum of two terms: a Ginzburg-Landau and a perimeter term. Our scaling allows low-energy states to be described by an order parameter which can have finitely many point singularities (vortex-like defects) of (possibly) fractional-degree connected by line discontinuities (string defects) of finite length. Our main result is a compactness and Γ-convergence theorem which shows how the coarse grained limit energy depends on the geometry of the manifold in driving the interaction between vortices and string defects. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Tangent vector fields of bounded variation 6 2.2. Vorticity 12 3. Statement of the main result 18 4. Proof of Γ-Convergence 20 4.1. Compactness 20 4.2. Γ-liminf 37 4.3. Γ-limsup 45 Appendix 49 Appendix A. Proof of the decomposition theorem 49 Appendix B. Proof of the ball construction in open subsets 60 References 62