Weingarten Surfaces Associated to Laguerre Minimal Surfaces

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 Σ, which depends on the principal curvatures of Σ 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.