Darboux transforms of a harmonic inverse mean curvature surface

Abstract

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward B\"{a}cklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward B\"{a}cklund transform. For a given isothermic harmonic inverse mean curvature surface, its classical Darboux transform is a harmonic inverse mean curvature surface. Then a transform of a solution to the Painlev\'{e} III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution.