# Interaction energy between vortices of vector fields on Riemannian surfaces

## TL;DR

This paper analyzes the interaction energy of vortices in tangent vector fields on Riemannian surfaces using a variational Ginzburg-Landau model, deriving a precise limit as the core size shrinks.

## Contribution

It provides a detailed second-order $\Gamma$-limit characterization of vortex interaction energy on Riemannian surfaces, extending previous models to vector fields with singularities.

## Key findings

- Interaction energy characterized as $\Gamma$-limit
- Vortices correspond to singular points with non-zero index
- Results applicable to tangent vector fields on 2D surfaces

## Abstract

We study a variational Ginzburg-Landau type model depending on a small parameter $\epsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $\epsilon\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $\Gamma$-limit (at the second order) as $\epsilon\to 0$.