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 \(H_S\) for any surface \(\Sigma \) , which depends on the principal curvatures of \(\Sigma \) and the radius function R. We then explore two classes of surfaces: those with \(H_S = 0\) , referred to as \(H_1\) -surfaces, and the surfaces with spherical mean curvature of harmonic type, denoted as \(H_2\) -surfaces. We provide a Weierstrass-type representation for the \(H_1\) -surfaces depending on two holomorphic functions, and a Weierstrass-type representation for the \(H_2\) -surfaces depending on three holomorphic functions. We prove that the \(H_1\) -surfaces are associated to minimal surfaces, whereas the \(H_2\) -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 \(H_1\) -surface/minimal surface classes or in \(H_2\) -surface/Laguerre minimal surface classes. We provide several examples and identify interesting minimal surfaces using our new Weierstrass-type representation.
