Infinitesimal Darboux transformations of the spectral curves of tori in the four-spaceстатья

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[1] Grinevich P. G., Taimanov I. A. Infinitesimal darboux transformations of the spectral curves of tori in the four-space // International Mathematics Research Notices. — 2007. — Vol. 2007. The present paper resumes the study of relations between the geometric properties of surfaces in R^3 and R^4 and the spectral properties of the corresponding Dirac operators. In the paper we study the behavior of the spectral curve of a torus in R^4 under conformal transformations of R^4 and, in particular, prove that the conformal transformations of R^4 which map a torus T∈R^4 into a compact torus preserve all Floquet multipliers of the corresponding Dirac operator. This generalizes the analogous result for tori in R^3 which was proved by M.U. Schmidt and the first author (P.G.G.), and confirmed the conjecture of the second author (I.A.T.) on the conformal invariance of the spectral curves of tori in R^3. Therewith by the spectral curve it was understood the analytic set M(Γ) in C^2 formed by the Floquet multipliers on the zero energy level. [ DOI ]

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