Weingarten Surfaces Associated to Laguerre Minimal Surfaces

TL;DR

This paper introduces new Weierstrass-type representations linking Laguerre minimal surfaces to other surface classes via holomorphic functions, expanding the understanding of their geometric properties and providing explicit examples.

Contribution

It defines spherical mean curvature for surfaces, classifies surfaces into $H_1$ and $H_2$ types, and establishes new representations connecting Laguerre minimal and minimal surfaces.

Findings

$H_1$-surfaces are associated with minimal surfaces.

$H_2$-surfaces are related to Laguerre minimal surfaces.

A new Weierstrass-type representation for Laguerre minimal surfaces is provided.

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.