Schroedinger flow into almost Hermitian manifolds

Details Schroedinger flow into almost Hermitian manifolds Schroedinger flow into almost Hermitian manifolds

2008-07-22

arxiv.org/abs/0807.3395v2

Mathematics

Analysis of PDEs

13 pages, no figure, minor change

Scientific paper

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.

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