A purely analytic derivation of Bonnet surfaces

TL;DR

The work provides a purely analytic derivation of Bonnet surfaces by enforcing a single-variable mean curvature $H=h(\xi)$ and the Painlevé property on the resulting ODE, which fixes the coefficient structure and identifies the equation with the Bonnet-type third-order ODE. The Painlevé test links the integral of motion to the logarithmic derivative of a Painlevé tau-function for $P_{VI}$ (codimension two) or $P_V$ in the limit, revealing a deep connection between differential geometry and integrable systems. The reduced Gauss-Codazzi data yield explicit expressions for the Hopf differential $Q$ and metric factor $\upsilon$, showing consistency with Bonnet's original surface family up to conformal transformations. The paper demonstrates a general analytic strategy for geometry problems with a single dominant variable, bridging classical differential geometry with Painlevé transcendents and tau-functions.

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 Painlevé property.