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In this paper we consider the gradient flow of the following Ginzburg-Landautype 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 andoriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative)and depends on a small parameter $\varepsilon>0$. If the energy satisfiesproper bounds, when $\varepsilon\to 0$ the second term forces the vector fieldsto have unit length. However, due to the incompatibility for vector fields on$M$ between the Sobolev regularity and the unit norm constraint, criticalpoints of $F_\varepsilon$ tend to generate a finite number of singular points(called vortices) having non-zero index (when the Euler characteristic isnon-zero). These types of problems have been extensively analyzed in a recentpaper by R. Ignat and R. Jerrard. As in Euclidean case, the position of thevortices is ruled by the so-called renormalized energy.  In this paper we are interested in the dynamics of vortices. We rigorouslyprove that the vortices move according to the gradient flow of the renormalizedenergy, which is the limit behavior when $\varepsilon\to 0$ of the gradientflow of the Ginzburg-Landau energy.

Dynamics of Ginzburg-Landau vortices for vector fields on surfaces