Elementary Conformal Surface Parametrizations = Parametrizaciones conformes elementales de superficies

Resumen

The first main result is the following theorem: Given a conformal parametrization of a 2-dimensional surface in R^n whose component functions are all polynomial in the parameters, it must be harmonic. As a first corollary, every surface in R^n that admits a conformal polynomial parametrization must be a minimal surface. The second main result consists of interesting explicit examples of rational conformal parametrizations defining surfaces in R^3 that are not Willmore surfaces. The resulting surfaces are thus neither minimal nor inversions of minimal surfaces. The main tool for this construction is the spinorial surface representation. In addition to the non-Willmore rational examples, some other non-trivial examples of explicit conformal parametrizations are obtained using this method. A third result establishes rigidity of conformal polynomial parametrizations of m-dimensional submanifolds, with m>=3, in the euclidean space R^n: The only conformal polynomial immersions of R^m into R^n, with n > m >=3, are the affine ones. The surface must be an m-plane.