Reductions of Gauss-Codazzi equations

TL;DR

This paper demonstrates that conformally parametrized surfaces in Euclidean space with constant curvature can be reduced via symmetry to solutions involving the sixth Painlevé function, unifying known solutions.

Contribution

It introduces a symmetry reduction method for Gauss-Codazzi equations that encompasses all known solutions expressed through the sixth Painlevé function.

Findings

General solution expressed with sixth Painlevé function

Recovery of known Bonnet and Bobenko-Eitner-Kitaev solutions

Unified framework for solutions of Gauss-Codazzi equations

Abstract

We prove that conformally parametrized surfaces in Euclidean space $\Rcubec$ of curvature $c$ admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painlev\'e function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner and Kitaev 1997) can be recovered by such a reduction.