Surfaces in the four-space and the Davey--Stewartson equations

Details Surfaces in the four-space and the Davey--Stewartson equations Surfaces in the four-space and the Davey--Stewartson equations

2004-01-29

arxiv.org/abs/math/0401412v1

J. of Geometry and Physics 56 (2006), 1235-1256

Mathematics

Differential Geometry

24 pages

Scientific paper

10.1016/j.geomphys.2005.06.013

We show that any equation from the Davey--Stewartson hierarchy induces an infinite family of geometrically different deformations of tori in $\R^4$ preserving the Willmore functional. We expose a derivation of the Weierstrass representation for surfaces in the four-space which is not unique in difference from the case of surfaces in the three-space. This non-uniqueness implies that the spectral curve of a torus in $\R^4$ is not uniquely defined as a complex curve formed by the Floquet multipliers.

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