Geometry

Volume 2013 (2013), Article ID 902092, 9 pages

http://dx.doi.org/10.1155/2013/902092

## Darboux Transforms of a Harmonic Inverse Mean Curvature Surface

Division of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,
Ibaraki 305-8571, Japan

Received 18 December 2012; Accepted 17 February 2013

Academic Editor: Manuel Sanchis

Copyright © 2013 Katsuhiro Moriya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.