Darboux Transforms of a Harmonic Inverse Mean Curvature Surface

Katsuhiro Moriya

doi:10.1155/2013/902092

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.

Highlights

The theory of surfaces is connected with the theory of solitons through a compatibility condition of the Gauss-Weingarten equations
A quaternionic line trivial bundle with a complex structure over a Riemann surface is associated with a conformal map from the Riemann surface to E4
All Darboux transforms of a conformal map describe a multiplier spectral curve of a conformal map

Summary

Introduction

The theory of surfaces is connected with the theory of solitons through a compatibility condition of the Gauss-Weingarten equations. We define a counterpart of a backward Backlund transform of a Willmore surface in a GHIMC surface as follows. All HIMC surfaces of revolution are classified by the solutions to the Painleve III equation in trigonometric form (42) (see Theorem 20). Let φ(x) be a solution to the Painleve III equation in trigonometric form (42) with φ󸀠(x) + 2 sin(φ(x)) ≠ 0, and let f : {x + yi ∈ C | x > 0} → Im H be an HIMC surface of revolution defined by Theorem 20. ̂f is an HIMC surface of revolution invariant under R, and the function φ(x) defined by (41) forf is a solution to the Painleve III equation in trigonometric form (42)

Conformal Maps

Transforms

GHIMC Surfaces