*Accepted Paper*

**Inserted:** 2 aug 2022

**Last Updated:** 3 aug 2022

**Journal:** Discrete Contin. Dyn. Syst. Ser. S

**Year:** 2021

**Abstract:**

We consider the gradient flow of a Ginzburg-Landau functional of the type \[
F_\varepsilon^{\mathrm{extr}}(u):=\frac{1}{2}\int_M \left

D u\right

_g^2 +
\left

\mathscr{S} u\right

^2_g
+\frac{1}{2\varepsilon^2}\left(\left

u\right

^2_g-1\right)^2\mathrm{vol}_g \]
which is defined for tangent vector fields (here $D$ stands for the covariant
derivative) on a closed surface $M\subseteq\mathbb{R}^3$ and includes extrinsic
effects via the shape operator $\mathscr{S}$ induced by the Euclidean embedding
of~$M$. The functional depends on the small parameter $\varepsilon>0$. When
$\varepsilon$ is small it is clear from the structure of the Ginzburg-Landau
functional that $\left

u\right

_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 $\varepsilon$ 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 7. In this paper we are interested the dynamics of vortices generated
by $F_\varepsilon^{\mathrm{extr}}$. To this end we study the behavior when
$\varepsilon\to 0$ of the solutions of the (properly rescaled) gradient flow of
$F_\varepsilon^{\mathrm{extr}}$. In the limit $\varepsilon\to 0$ we obtain the
effective dynamics of the vortices. The dynamics, as expected, is influenced by
both the intrinsic and extrinsic properties of the surface
$M\subseteq\mathbb{R}^3$.