Channel linear Weingarten surfaces in smooth and discrete differential geometry

Abstract

We consider channel linear Weingarten surfaces in spaceforms of arbitrary curvature and their discretisation in the sense of integrable discrete differential geometry. We use Lie sphere geometry to unify the different ambient spaces.In the smooth category, we prove that all non-tubular channel linear Weingarten surfaces are rotational within their ambient spaceforms. Further we give explicit parametrisations of all rotational surfaces with constant Gauss curvature in terms of Jacobi elliptic functions. Thus, we obtain a transparent classification of all channel linear Weingarten surfaces in spaceforms with non-negative curvature as well as a large subclass of channel linear Weingarten surfaces in hyperbolic spaceforms up to parallel transformation.We further point out how similar arguments to the ones presented in this text lead to explicit parametrisations of constant mean curvature surfaces.In the discrete category, we advance the theory of the recently defined class of discrete channel surfaces. Further, we prove that all strongly non-tubular discrete channel linear Weingarten surfaces are rotational within their ambient spaceforms. Also we outline how this will lead to a classification of large classes of channel linear Weingarten surfaces, similar to the results of the smooth theory.