Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Abstract In the work Laredo, 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 HS for any surface, which depends on the principal curvatures of and the radius function R. We then explore two classes of surfaces: those with HS = 0, referred to as H₁-surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as H₂-surfaces. We provide a Weierstrass-type representation for each of these classes depending on three holomorphic functions. We prove that the H₁-surfaces are associated to the minimal surfaces, whereas the H₂-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 H₁-surface/minimal surface classes or in H₂-surface/Laguerre minimal surface classes. We also characterize the rotational cases, allowing us to find a complete rotational Laguerre minimal surface.