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Schrödinger flow into almost Hermitian manifolds
Author(s) -
Chihara Hiroyuki
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds060
Subject(s) - mathematics , covariant derivative , hermitian manifold , manifold (fluid mechanics) , hermitian matrix , bounded function , connection (principal bundle) , riemannian manifold , pure mathematics , pullback , covariant transformation , operator (biology) , mathematical analysis , flow (mathematics) , mathematical physics , ricci curvature , geometry , curvature , mechanical engineering , biochemistry , chemistry , repressor , transcription factor , gene , engineering
We present a short‐time existence theorem of solutions to the initial value problem for Schrödinger 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 Schrödinger map equation.