arXiv subjects
Publications and source records attributed to Iskander A. Taimanov.
We describe the Ozawa solution to the Davey--Stewartson II equation from the point of view of surface theory by presenting a soliton deformation of surfaces which is ruled by the Ozawa solution. The Ozawa solution blows up at certain moment and we describe explicitly the corresponding singularity of the deformed surface.
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
In sections 1 and 2 we follow our online talk at the 21st Geometrical Seminar (Beograd, Serbia) on June 30, 2022 by giving a survey of the formality problem for manifold with special holonomy and exposing recent results by M. Amann and the author on the formality of Joyce's examples of G_2-manifolds. In section 3 we expose the approach to establishing the formality by using the intersection Massey products.
We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level.
This article surveys the Weierstrass representation of surfaces in the three- and four-dimensional spaces, with an emphasis on its relation to the Willmore functional. We also describe an application of this representation to constructing a new type of solutions to the Davey-Stewartson II equation. They have regular initial data, gain one-point singularities at certain moments of time, and extend to smooth solutions for the remaining times.
The Moutard transform is constructed for the solutions of the Davey-Stewartson II equation. It is geometrically interpreted using the spinor (Weierstrass) representation of surfaces in four-dimensional Euclidean space. Using the Moutard transformation and minimal surfaces, examples of solutions are constructed that have smooth, rapidly decreasing initial data and lose their regularity in a finite time.
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere.
The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. In this paper, we give a survey of basic notions and results related with the Guillemin-Uribe trace formula and provide concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example.
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simply-connected manifold which is not homeomorphic to a sphere.
We construct a canonical basis of two-cycles, on a $K3$ surface, in which the intersection form takes the canonical form $2E_8(-1) \oplus 3H$. The basic elements are realized by formal sums of smooth submanifolds.
We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range $(e_0,e_1)$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, $e_0=0$ is the minimal energy of the system).
We present an explicit example of a fast decaying solution to the modified Novikov--Veselov equation with a one-point singularity in the space-time. It is constructed by using the geometrical interpretation of the Moutard transformation of solutions to this equation and the Enneper minimal surface.
A construction of blowing up solutions to the modified Novikov-Veselov equation is proposed. It is based on the Moutard transformation of two-dimensional Dirac operators and its geometrical interpretation via surface geometry. An explicit example of such a solution constructed by using the Enneper minimal surface is discussed in detail. This example was briefly announced in arXiv:1408.4723.
We describe the action of the (Mobius) inversion on the data of the Weierstrass representation of surfaces in the three-space and show that the Moutard transformation of two-dimensional Dirac operators has a geometrical meaning: it maps the potential $U$ of a surface $S$ into the potential of its inversion.
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lamé potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, Riemann's "staircase" minimal surfaces appear as other examples related to 1--gap Lamé potentials.
We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves coincide with the spectral curves of certain elliptic KP solitons as studied by Krichever.