Darboux Transforms of a Harmonic Inverse Mean Curvature Surface

Abstract EN

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced.

A backward Bäcklund 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 Bäcklund 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 Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution.