Surfaces with harmonic inverse mean curvature and Painlevé equations
The authors study surfaces immersed in \(\mathbb{R}^3\) whose mean curvature \(H\) satisfies \(\Delta (1/H)=0\) where \(\Delta\) denotes the Laplacian on the surface. They call these surfaces harmonic inverse mean curvature surfaces or HIMC surfaces. They were first introduced by \textit{A. Bobenko} [in: `Harmonic maps and integrable systems', Aspects Math. E23, 83-127 (1994; Zbl 0841.53003)] in the context of the theory of integrable systems; they were further studied in [\textit{A. S. Fokas} and \textit{I. M. Gelfand}, Commun. Math. Phys. 177, 203-220 (1996; Zbl 0864.53003)] and [\textit{D. A. Korotkin}, `On some integrable cases in surface theory' (SFB 288 Preprint 151, TU Berlin) (1994)]. In this paper, the authors construct families of examples of HIMC surfaces which are not of constant mean curvature. They describe rotational HIMC surfaces and, more generally, a certain class of isothermic HIMC surfaces in terms of third resp. fifth and sixth Painlevé transcendents. They also study `Cartan cones', i.e., exceptional families of HIMC surfaces which do not stem from solutions of Painlevé equations. The representation of the Hopf differential of the studied subclass of isothermic surfaces as the Hamiltonian for Painlevé equations is fully explained in the Appendix of the paper.
harmonic inverse mean curvature surfaces
scientific article; zbMATH DE number 1098483