Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces.

We consider the gradient flow of a Ginzburg-Landau functional of the type \[ F_\eps^{\extr}(u):=\frac{1}{2}\int_M \abs{D u}_g^2 + \abs{\Sh u}^2_g +\frac{1}{2\eps^2}\left(\abs{u}^2_g-1\right)^2\Vg \] which is defined for tangent vector fields (here $D$ stands for the covariant derivative) on a closed surface~$M\subseteq\R^3$ and includes extrinsic effects via the shape operator $\Sh$ induced by the Euclidean embedding of~$M$. The functional depends on the small parameter $\eps>0$. When $\eps$ is small it is clear from the structure of the Ginzburg-Landau functional that $\abs{u}_g$ ``prefers'' to be close to $1$. However, due to the incompatibility for vector fields on $M$ between the Sobolev regularity and the unit norm constraint, when $\eps$ is close to $0$, it is expected that a finite number of singular points (called vortices) having non-zero index emerges (when the Euler characteristic is non-zero). This intuitive picture has been made precise in the recent work by R. Ignat \& R. Jerrard \cite{JerrardIgnat_full}. In this paper we are interested the dynamics of vortices generated by $F_\eps^{\extr}$. To this end we study the behavior when $\eps\to 0$ of the solutions of the (properly rescaled) gradient flow of $F_\eps^{\extr}$. In the limit $\eps\to 0$ we obtain the effective dynamics of the vortices. The dynamics, as expected, is influenced by both the intrinsic and extrinsic properties of the surface~$M\subseteq\R^3$.