Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair
In the first part of this paper the author reworks the Gauss-Weingarten equations via the moving frames for a Riemann surface parametrized by means of the conformal parameters \((z, \bar z).\) Using suitable compatibility conditions, the Gauss-Codazzi equations are derived in terms of the conformal factor of the metric \(e^u,\) the mean curvature \(H,\) and the quantity \(Q = \langle F_{zz}, N \rangle ,\) with \(N\) being the unit normal and \(F\) the (parametrized) position vector in \(\mathbb R^3\). In the second part of the paper the author establishes a connection between quaternions and surfaces in \(\mathbb R^3,\) which gives rise to a certain identification of the Gauss-Weingarten equations with the Lax representation for Painlevé equations by means of an \(SU(2)\) transformation. As an example, the author considers a generalized Weierstraß representation for a constant-mean-curvature hypersurface and the Enneper minimal surface as special cases of solutions of the above pairing. Several typos, ambiguities, and reliance on other papers make the exposition hard to follow.
scientific article; zbMATH DE number 2168642