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Painlevé equations in the differential geometry of surfaces. (English) Zbl 0970.53002
Lecture Notes in Mathematics. 1753. Berlin: Springer. vi, 120 p. (2000).
Painlevé equations are nonlinear ordinary differential equations of second order such that the solutions may have singularities other than poles only at certain fixed points (the Painlevé property). Alternatively, these equations can be introduced as equations of isomonodromic deformations: they arise from the compatibility condition $$A_t-W_\lambda+[A,W]=0$$ for a system $$\psi_\lambda =A\psi$$, $$\psi_t= W\psi$$ (variable $$t$$, parameter $$\lambda$$, $$A$$ and $$W$$ are rational functions of $$\lambda)$$. On the other hand, the classical Gauss-Weingarten equations $$\Phi_x=U$$, $$\Phi_y=V$$ for the moving frame $$\Phi(x,y)$$ of a surface $$(x,y)\mapsto z(x,y)$$ in $$\mathbb{R}^3$$ lead to the formally analogous compatibility condition $$U_x-V_y +[U,V]=0$$. The latter condition for (matrix-valued) functions $$U,V$$ is said to be integrable if a solution $$U=U(x,y, \lambda)$$, $$V=V(x,y, \lambda)$$ depending on a (spectral) parameter $$\lambda$$ exists (the Zakharov-Shabad representation). It follows that the Painlevé property is closely related to certain special classes of surfaces (better: to the soliton theory). The goal of this book is to describe gometric properties of surfaces leading to Painlevé equations, use the theory of Painlevé equations to study the global geometry of the surfaces and, conversely, treat the geometrically motivated problems for Painlevé equations. Most of the book is devoted to Bonnet surfaces possessing one-parameter families of isometries preserving the mean curvature function (we may refer to the authors’ article [J. reine angew. Math. 499, 47-79 (1998; Zbl 0906.53001)]). The results are subsequently adapted for the elliptic and hyperbolic space forms. Quite other classes of surfaces are treated in the second half of the book: the surfaces with constant negative curvature and two straight asymptotic lines (the Amsler surfaces), the constant mean curvature surfaces with internal isometry (the Smyth surfaces), and an analogue of the Amsler surfaces in affine differential geometry. The moving frames complemented by quaternionic techniques are employed. The book involves many geometrical results in the best classical sense. They are illustrated with beautiful pictures and provide a nice contribution both to differential geometry and to the theory of Painlevé equations.

##### MSC:
 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 53A05 Surfaces in Euclidean and related spaces 53A15 Affine differential geometry 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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