Schroedinger flow into almost Hermitian manifolds

TL;DR

This paper proves short-time existence of solutions for Schroedinger maps from a closed Riemannian manifold to an almost Hermitian manifold, overcoming derivative loss issues with a pseudodifferential operator.

Contribution

Introduces a pseudodifferential operator technique to handle derivative loss in Schroedinger maps into almost Hermitian manifolds, establishing a short-time existence theorem.

Findings

Established short-time existence for Schroedinger maps into almost Hermitian manifolds.

Developed a pseudodifferential operator method to address non-parallel almost complex structures.

Overcame derivative loss obstacle in the energy method for this problem.

Abstract

We present a short-time existence theorem of solutions to the initial value problem for Schroedinger maps of a closed Riemannian manifold to a compact almost Hermitian manifold. 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 essentially eliminate the loss of one derivative from the partial differential equation of the Schroedinger map.