A third order dispersive flow for closed curves into almost Hermitian manifolds

TL;DR

This paper proves a short-time existence theorem for a third order dispersive flow of closed curves into almost Hermitian manifolds, addressing analytical challenges posed by non-parallel almost complex structures.

Contribution

It introduces a novel pseudodifferential operator technique to handle derivative loss caused by non-parallel almost complex structures in dispersive flows.

Findings

Established a short-time existence result for the dispersive flow.

Developed a pseudodifferential operator method to manage derivative loss.

Extended the analysis of vortex filament models to almost Hermitian manifolds.

Abstract

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow.