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 typeF epsilon extr(u) : 1/2 integral(M) vertical bar Du vertical bar(2)(g) + vertical bar pu vertical bar(2)(g) + 1/2 epsilon(2) (vertical bar u vertical bar(2)(g) - 1 )(2) vol(g)which is defined for tangent vector fields (here D stands for the covariant derivative) on a closed surface M subset of R-3 and includes extrinsic effects via the shape operator p induced by the Euclidean embedding of M. The functional depends on the small parameter epsilon > 0. When epsilon is small it is clear from the structure of the Ginzburg-Landau functional that vertical bar u vertical bar(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 epsilon is close to 0, it is expected that a finite number of singular points (called vortices) having nonzero 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 H. In this paper we are interested the dynamics of vortices generated by F-epsilon(extr). To this end we study the behavior when epsilon -> 0 of the solutions of the (properly rescaled) gradient flow of F-epsilon(extr). In the limit epsilon -> 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 subset of R-3.