Generalized Weierstrass–Enneper representations of Euclidean, spacelike, and timelike surfaces: a unified Lie-algebraic formulation

Abstract

The complex-analytic approach to constructing minimal surfaces carried out by Weierstrass and Enneper has since been extended to create conformal parametrizations of minimal and more general surfaces, including Euclidean, spacelike, and timelike surfaces in three-dimensional Euclidean and Lorentz spaces. In this work, we present a Lie-algebraic formulation that unites these representations into one coherent framework and completes the formulas in the general timelike case. Integrable moving frames for surfaces are created inside of three-dimensional Lie algebras by applying a scaling, together with an isometric action of the appropriate Lie group, to a standard constant orthonormal frame. Using complex coordinates for Euclidean and spacelike surfaces and hyperbolic coordinates for timelike surfaces allows for simple characterizations of geometric properties of the surfaces in terms of the complex and hyperbolic structures. We give expressions in this framework of the first and second fundamental forms and of mean and Gaussian curvatures on Euclidean, spacelike, and timelike surfaces. The expression of the Gaussian curvature affirms in an efficient and elegant way that Gauss’ Theorem Egregium holds for surfaces of all three types. We also provide examples and illustrations of special surfaces and deformations of surfaces arising from the Weierstrass–Enneper representations.