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Sfb 288 Differential Geometry and Quantum Physics |
Painleve Functions in Differential Geometry
Some Applications in Classical Surface Theory
Painleve transcendents (or just Painleve functions) are the solutions of a particular set of non-linear second order ordinary differential equation. They define new transcendental functions.
Painleve transcendents are used in many apllications in physics and chemistry. It is also known that CMC surfaces with rotational symmetric metric as well as Amsler surfaces can be decribed in terms of Painleve (namely Painleve III) functions. In 1994 it was shown by Bobenko and Eitner that the classical problem of Bonnet surfaces (surfaces that admits one-parametric families of isometries preserving the principle curvature) can be solved in terms of those functions. In the earlier 90th A.Bobenko discovered (or at least re-discovered) a case of surface whose reciprocal of the mean curvature is a harmonic map with respect to the induced metric (HarmonicReciprocalMeanCurvature surfaces). All isothermic surfaces of this type are dual (generated by a sphere-congruence) to Bonnet surfaces and can therefore be described in Painleve functions, too. In a recent paper Bobenko, Eitner and Kitaev defined "Theta-isothermic" surfaces. They admits also dual surfaces, no longer in an Euclidean space but in the 3-dimensional sphere or hyperbolic space. All "Theta-isothermic" HRMC surface are found: they can be parametrized in terms of Painleve functions. Moreover their dual surfaces are all Bonnet surfaces in S3 and H3, respectively.
Other integral systems have self-similarity solutions that can be expressed in terms of Painleve functions, like the Tzitzeica equation for the Blaschke-metric of affine spheres. Willmore surfaces with rotational symmetric metric of the conformal Gauss map, Bianchi-surfaces coming from the deformation of a surface of revolution or a helicoidal surface.
However, not in any case the geometrical meanings of this self-similiarity solutions is obvious.
At least the following applications for Painleve functions in surface theory are meaningful:
Bonnet surfaces in 3-dimensional euclidean space
"theta-isothermic" HRMC and their dual surfaces
Bianchi surfaces (deformation of a surface of revolution or a helicoidal surface). In general the surfaces of these types are rather different but surprisingly the family acts in all cases similiar: it preserves the ratio of the principle curvatures. This is true for the CMC surfaces with rotational symmetrical metric as well as for the Amsler surfaces (a subset of Bianchi surfaces). However, while the family-parameter in the latter cases is isospectral, in the further it is non-isospectral.
Author: Ulrich Eitner