arXiv subjects

Alexander I. Bobenko

Publications and source records attributed to Alexander I. Bobenko.

60 records · Page 4Linked to original sources

arXiv2001

Hexagonal Circle Patterns and Integrable Systems. Patterns with Constant Angles

Hexagonal circle patterns with constant intersection angles are introduced and studied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings $z^c$ and $\log z$ are constructed as special isomonodromic solutions. Circle patterns studied in the paper include Schramm's circle patterns with the combinatorics of the square grid as a special case.

math.CV↗

arXiv2000

Conformally symmetric circle packings. A generalization of Doyle spirals

From the geometric study of the elementary cell of hexagonal circle packings --- a flower of 7 circles --- the class of conformally symmetric circle packings is defined. Up to Moebius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are given explicitly. It is shown, that these circle packings can be viewed as discretizations of the quotient of two Airy functions. There is an interactive version of this paper that contains JAVA applets that let you experiment with the circle packings directly. It is available at http://www-sfb288.math.tu-berlin.de/Publications/online/

math.CV↗

arXiv1999

Exploring Surfaces through Methods from the Theory of Integrable Systems. lectures on the Bonnet problem

A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known as the Bonnet problem. Regarding the Bonnet problem, we show how analytic methods of the theory of integrable systems -- such as finite-gap integration, isomonodromic deformation, and loop group description -- can be applied for studying global properties of special surfaces. This paper presents the contents of the lectures given at the School on Differential Geometry on 12-30 April 1999 at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste.

math.DG↗

arXiv1999

Discrete Z^a and Painleve equations

A discrete analogue of the holomorphic map z^a is studied. It is given by a Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding immersed circle patterns lead to special separatrix solutions of a discrete Painleve equation. Global properties of these solutions, as well as of the discrete $z^a$ are established.

solv-int↗

arXiv1999

Discrete Lagrangian reduction, discrete Euler-Poincare equations, and semidirect products

A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of $G$. In this context the reduction of the discrete Euler-Lagrange equations is shown to lead to the so called discrete Euler-Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler-Poincaré equations leads to discrete Hamiltonian (Lie-Poisson) systems on a dual space to a semiproduct Lie algebra.

math.SG↗

arXiv1998

Orthogonal nets and Clifford algebras

A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is discussed in a conformal setting. This way, the notions of ``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal systems'' are obtained --- the latter as a generalization of discrete orthogonal systems in Euclidean space. The relation of a Cauchy problem for discrete orthogonal nets and a permutability theorem for the Ribaucour transformation of smooth orthogonal systems is discussed.

math.DG↗