Surfaces with Harmonic Inverse Mean Curvature and Painlev Equations

In this paper we study surfaces immersed in R such that the mean curvature function H satisfies the equation (1=H) = 0, where 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 Painlevé transcendent. In the general class of HIMC surfaces we distinguish a subclass of -isothermic surfaces, which is a generalization of the isothermic HIMC surfaces, and classify all the -isothermic HIMC surfaces in terms of the solutions of the fifth and sixth Painlevé transcendents. Mathematics Subject Classifications (1991): 53A10, 53C42, 53A05, 34A12, 34E05, 35Q58.