Weingarten Surfaces Associated to Laguerre Minimal Surfaces

Abstract

In the work \cite{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 for any surface $\Sigma$ its spherical mean curvature $H_S$ which depends on principal curvatures of $\Sigma$ and the radius function $R$. Then we consider two classes of surfaces: the ones with $H_S = 0$, called $H_1$-surfaces, and the surfaces with spherical mean curvature of harmonic type, named $H_2$-surfaces. We provide for each these classes a Weierstrass-type representation depending on three holomorphic functions and we prove that the $H_1$-surfaces are associated to the minimal surfaces, whereas the $H_2$-surfaces are related to the Laguerre minimal surfaces. As application we provide a new Weierstrass-type representation for the Laguerre minimal surfaces - and in particular for the minimal surfaces - in such a way that the same holomorphic data provide examples in $H_1$-surface/minimal surface classes or in $H_2$-surface/Laguerre minimal surface classes. We also characterize the rotational cases, what allow us finding a complete rotational Laguerre minimal surface.