A purely analytic derivation of Bonnet surfaces

TL;DR

This paper analytically characterizes Bonnet surfaces by reducing their mean curvature to an ODE with the Painlevé property, providing a purely analytic approach to understanding these geometric surfaces.

Contribution

It introduces a novel analytic method to characterize Bonnet surfaces through differential equations and the Painlevé property, bypassing geometric conditions.

Findings

Mean curvature reduction to an ODE

ODE possesses the Painlevé property

Analytic characterization of Bonnet surfaces

Abstract

Bonnet has characterized his surfaces by a geometric condition. What is done here is a characterization of the same surfaces by two analytic conditions: (i) the mean curvature $H$ of a surface in $\mathbb{R}^3$ should admit a reduction to an ordinary differential equation; (ii) this latter equation should possess the Painlev\'e property.