Conformal parametrizations defined by polynomials

Abstract

The current paper discusses some new results about conformal polynomial surface parametrizations. A new theorem is proved: Given a conformal polynomial surface parametrization of any degree it must be harmonic on each component.As a first geometrical application, the next theorem is shown: Every surface that admits a conformal polynomial parametrization must be a minimal surface. This is not the case for rational conformal polynomial parametrizations, where the conformal condition doesn't imply that polynomial components must be harmonic.A new general theorem is established for conformal polynomial parametrizations of m-dimensional surfaces on the euclidean space Rn: The only conformal polynomial parametrizations of a m-dimensional surfaces, in Rn, with m>2 and n≥m, must be formed by linear polynomials, i.e. the surface parametrization must be an affine transformation of the usual cartesian framework.