Instability of Ginzburg-Landau Vortices on Manifolds

TL;DR

This paper studies the instability and annihilation of vortices in Ginzburg-Landau models on manifolds, revealing conditions under which vortices are unstable or tend to vanish, with implications for geometric analysis.

Contribution

It provides new results on vortex instability and annihilation in Ginzburg-Landau models on specific manifolds, extending understanding of vortex dynamics in geometric contexts.

Findings

Vortices are unstable on certain compact 2-manifolds.

Vortex annihilation occurs under heat flow on surfaces of revolution.

Results contribute to the understanding of vortex behavior in geometric settings.

Abstract

We investigate two settings of Ginzburg-Landau posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg-Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg-Landau heat flow posed on certain surfaces of revolution with boundary.