Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 151

Bonnet Surfaces and Painleve Equations

A. Bobenko, U. Eitner

J. reine angew. Math. 499 (1998) 47-79

A classical problem of the theory of surfaces in R^3 is whether there exists non-trivial smooth families of isometric surfaces with the same principle curvatures or not. Bonnet was able to show that beside CMC ( Constant Mean Curvature ) surfaces there is a class of surfaces depending of finitly many parameters which allows a smooth deformation preserving the principle curvature ( which is here the same as mean curvature). These surfaces are called Bonnet surfaces. In our preprint we show that Bonnet surfaces are describeable in "classical" Painleve transcendents. Since these functions are known as well as other classical functions, all Bonnet surfaces are now known explicitly. In some special cases such surfaces can be expressed in hypergeometric or elliptic functions. This is discussed in the appendices.

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