The Invariant Symplectic Action and Decay for Vortices

Abstract

The (local) invariant symplectic action functional $\A$ is associated to a Hamiltonian action of a compact connected Lie group $\G$ on a symplectic manifold $(M,\omega)$, endowed with a $\G$-invariant Riemannian metric $<\cdot,\cdot>_M$. It is defined on the set of pairs of loops $(x,\xi):S^1\to M\x\Lie\G$ for which $x$ satisfies some admissibility condition. I prove a sharp isoperimetric inequality for $\A$ if $<\cdot,\cdot>_M$ is induced by some $\omega$-compatible and $\G$-invariant almost complex structure $J$, and, as an application, an optimal result about the decay at $\infty$ of symplectic vortices on the half-cylinder $[0,\infty)\x S^1$.