A formula for constructing infinitely many surfaces on Lie algebras and integrable equations

Abstract

Surfaces immersed in Lie algebras can be characterized by the so called fundamental forms. The coefficients of these forms satisfy a system of nonlinear partial differential equations (PDEs), the Gauss–Mainardi–Codazzi–Ricci equations. For particular surfaces, this system of PDEs belongs to a distinguished class of equations known as integrable equations. Such an example in \( {\Bbb R}^3 \) is the class of surfaces of constant mean curvature which is associated with the sinh-Gordon equation. Here an explicit formula is presented which associates with a given system of integrable nonlinear PDEs infinitely many surfaces immersed in Lie algebras. This formula is based on the general construction of surfaces on Lie algebras introduced recently by the first two authors, and on the fact that integrable equations possess infinitely many symmetries. Several examples of surfaces immersed in the 3-dimensional Euclidean space are discussed, including the list of integrable surfaces recently presented by Bobenko and certain deformations thereof.