Abstract

We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data $u_0^\epsilon $ for which the jacobian $Ju_0^\epsilon $ concentrates around an rectifiable $m-2$ dimensional set, say $\Gamma _0$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t>0$ continues to concentrate around some codimension $2$ submanifold, say $\Gamma _t$, and that the family $\lbrace \Gamma _t \rbrace $ of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when $\Gamma _0$ is a round $m-2$-dimensional sphere with multiplicity $1$. We also prove a number of partial results for more general inital data