Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair (Q1774696)

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.