Darboux transforms of a harmonic inverse mean curvature surface

TL;DR

This paper introduces a new class of surfaces in four-dimensional space, defines their transformations, and connects these geometric transformations to solutions of the Painlevé III equation, expanding the understanding of harmonic inverse mean curvature surfaces.

Contribution

It defines generalized harmonic inverse mean curvature surfaces in 4D, constructs their Darboux transforms via backward Bäcklund transforms, and links these to solutions of Painlevé III.

Findings

Introduction of generalized harmonic inverse mean curvature surfaces in 4D

Construction of Darboux transforms using backward Bäcklund transforms

Connection between Darboux transforms and Painlevé III solutions

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.