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The present article is a sequel to [19, 20]. The results presented here extend onto general surfaces the results obtained in [20] for surfaces of revolution and were exposed in a lot of talks of the author during the last year being at the end announced in [21]. These are – a construction of a global Weierstrass representation for an arbitrary closed… (More)

In the present article we examine in details global deformations of surfaces of revolution via the modified Korteweg–de Vries (mKdV) equations and the first integrals, of these deformations, regarded as invariants of surfaces. It is a sequel to our paper [8] where the general case of modified Novikov–Veselov (mNV) deformations is considered. Since the main… (More)

It is shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass–Enneper inducing has Hamiltonian form. Its simplest finite–dimensional reduction is the integrable Hamiltonian system with two degrees of freedom. This finite-dimensional system admits S-action and classes of S-equivalence of its trajectories are… (More)

- Ivan K. BABENKO, Iskander A. TAIMANOV
- 1998

A smooth manifold M is called symplectic if it carries a nondegenerate closed 2-form ω which is called a symplectic form. In this event a symplectic manifold means a pair (M,ω). Since the skew-symmetric form ω is nondegenerate, M is even-dimensional and moreover such manifold always carries an almost complex structure. Topology of symplectic manifolds is… (More)

In the present article we consider Weierstrass representations of spheres in R. An existence of a global Weierstrass representation for any compact oriented surface of genus g ≥ 1 has been established in [19, 20] and this proof, in fact, works for spheres also. Being mostly interested in relations of these representations to the spectral theory and in… (More)

- Ivan K. BABENKO, Iskander A. TAIMANOV
- 1999

In this paper we resume our study of the formality problem for symplectic manifolds, which we started in [1, 2] where the first examples of nonformal simply connected symplectic compact manifolds were constructed and a method of constructing such manifolds by symplectic blow-ups was introduced. A smooth manifold X is called symplectic if there is a… (More)

The Weierstrass representation for surfaces in R3 [8, 13] was generalized for surfaces in R4 in [12] (see also [4]). This paper uses the quaternion language and the explicit formulas for such a representation were written by Konopelchenko in [9] for constructing surfaces which admit soliton deformation governed by the Davey–Stewartson equations. This… (More)

In this paper we show how to assign to any torus immersed into the threespace R or the unit three-sphere S a complex curve such that the immersion is described by functions defined on this curve (a Riemann surface which is generically of infinite genus). We call this curve the spectrum of a torus (with a fixed conformal parameter). This spectrum has many… (More)

It is known that for n ≥ 3 centres and positive energies the ncentre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary g > 1 independent integrals of Gevrey class g exist, no real-analytic (that is, Gevrey class 1)… (More)

- I A Taimanov
- 2006

(see Theorem 1 and Corollary 1). This generalizes the analogous result for tori in R which was proved by M.U. Schmidt and the first author (P.G.G.) [8] and confirmed the conjecture of the second author (I.A.T.) on the conformal invariance of the spectral curves of tori in R. Therewith by the spectral curve it was understood the analytic set M(Γ) in C formed… (More)