  In this paper we consider the gradient flow of the following Ginzburg-Landau
type energy \[
  F_\varepsilon(u) := \frac{1}{2}\int_{M}\vert D u\vert_g^2
+\frac{1}{2\varepsilon^2}\left(\vert u\vert_g^2-1\right)^2\mathrm{vol}_g. \]
This energy is defined on tangent vector fields on a $2$-dimensional closed and
oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative)
and depends on a small parameter $\varepsilon>0$. If the energy satisfies
proper bounds, when $\varepsilon\to 0$ the second term forces the vector fields
to have unit length. However, due to the incompatibility for vector fields on
$M$ between the Sobolev regularity and the unit norm constraint, critical
points of $F_\varepsilon$ tend to generate a finite number of singular points
(called vortices) having non-zero index (when the Euler characteristic is
non-zero). These types of problems have been extensively analyzed in a recent
paper by R. Ignat and R. Jerrard. As in Euclidean case, the position of the
vortices is ruled by the so-called renormalized energy.
  In this paper we are interested in the dynamics of vortices. We rigorously
prove that the vortices move according to the gradient flow of the renormalized
energy, which is the limit behavior when $\varepsilon\to 0$ of the gradient
flow of the Ginzburg-Landau energy.
