Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 182

Harmonic Inverse Mean Curvature Surfaces and Painleve Equations

A. Bobenko, U. Eitner, A.V. Kitaev

Geometriae Dedicata 68 (1997) 187-227

In this paper we study the surfaces immersed in R^3 such that the mean curvature function $H$ satisfies the equation D(1/H) = 0, where D is the Laplace operator of the induced metric. We call them HIMC surfaces. All HIMC surfaces of revolution are classified in terms of the third Painleve transcendent. In the general class of HIMC surfaces we distinguish a subclass of theta-isothermic surfaces, which is a generalization of the isothermic HIMC surfaces, and classify all the theta-isothermic HIMC surfaces in terms of the solutions of the fifth and sixth Painleve transcendents.

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