$k$-symmetric AKS systems and flat immersions in spheres

Details $k$-symmetric AKS systems and flat immersions in spheres $k$-symmetric AKS systems and flat immersions in spheres

2006-04-11

arxiv.org/abs/math/0604245v2

J. London Math. Soc. (2) 77 (2008) 299-319

Mathematics

Differential Geometry

21 pages. Minor typographical corrections

Scientific paper

10.1112/jlms/jdm109

We define a large class of integrable nonlinear PDE's, \emph{$k$-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing $n$-dimensional Euclidean space into a sphere of dimension $2n-1$ can be addressed via this scheme, producing infinitely many real analytic solutions.

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