Weingarten surfaces associated to Laguerre minimal surfaces

Abstract

In the work [20], the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function R is a geometric invariant of hypersurface. In this paper, we define the spherical mean curvature for any surface , which depends on the principal curvatures of and the radius function R. We then explore two classes of surfaces: those with , referred to as -surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as -surfaces. We provide a Weierstrass-type representation for the -surfaces depending on two holomorphic functions, and a Weierstrass-type representation for the -surfaces depending on three holomorphic functions. We prove that the -surfaces are associated to minimal surfaces, whereas the -surfaces are related to Laguerre minimal surfaces. As an application, we present a new Weierstrass-type representation for Laguerre minimal surfaces, and specifically for minimal surfaces. In this way, the same holomorphic data can be used to provide examples in -surface/minimal surface classes or in -surface/Laguerre minimal surface classes. We provide several examples and identify interesting minimal surfaces using our new Weierstrass-type representation.