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Iskander A. Taimanov

Publications and source records attributed to Iskander A. Taimanov.

27 records · Page 2Linked to original sources
arXiv2006

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see math.DG/0503707). By using surfaces of revolution we demonstrate that it resembles the Willmore functional for surfaces in the Euclidean space in many geometrical respects. We also observe the relation of these functionals to the isoperimetric problem.

math.DG↗
arXiv2006

Two-dimensional Dirac operator and surface theory

We give a survey on the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional and on related problems of spectral theory of the two-dimensional Dirac operator with periodic coefficients.

math.DG↗
arXiv2005

Surfaces in three-dimensional Lie groups

We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of the Willmore functional for surfaces in these geometries.

math.DG↗
arXiv2004

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

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.

math.DG↗
arXiv2004

On the integrability of the n-centre problem

It is known that for $n \geq 3$ centres and positive energies the $n$-centre 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) independent integral exists.

math.DS↗
arXiv1998

On nonformal simply connected symplectic manifolds

For any $N \geq 5$ nonformal simply connected symplectic manifolds of dimension $2N$ are constructed. This disproves the formality conjecture for simply connected symplectic manifolds which was introduced by Lupton and Oprea.

math.SG↗
arXiv1998

The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres

The Weierstrass representation for spheres in $\R^3$ and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or of a cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them a lower estimate of the Willmore functional in terms of the dimension of the kernel of the corresponding Dirac operator on a two-sphere is obtained. It is conjectured that this estimate is valid for all Dirac operators on spheres and some reasonings for this conjecture are discussed. In Appendix a criterion distinguishing Weierstrass representations, of universal coverings of compact surfaces of higher genera, converted into immersions of compact surfaces is given.

math.DG↗
arXiv1997

The Weierstrass representation of closed surfaces in $R^3$

In the present paper which a sequel to dg-ga/9511005 and dg-ga//9610013 a global Weierstrass representation of an arbitrary closed oriented surface of genus $\geq 1$ in the the three-space is constructed. The Weierstrass spectrum of a torus immersed into $R^3$ is introduced and finite-zone planes as well as finite-zone solutions to the modified Novikov-Veselov equations are constructed.

dg-ga↗