2022/09/28 by Laredo Rennan Pereira Santos, Santos, Laredo Rennan Pereira, Armando M. V. Corro +1
Engineering · Mathematics · #30F15 #53A05 #53A07 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #G.0 #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2209.14336
openalex publication_date 2022/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the work \citeLaredo 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 Σ its spherical mean curvature HS which depends on principal curvatures of Σ and the radius function R. Then we consider two classes of surfaces: the ones with HS = 0, called H1-surfaces, and the surfaces with spherical mean curvature of harmonic type, named H2-surfaces. We provide for each these classes a Weierstrass-type representation depending on three holomorphic functions and we prove that the H1-surfaces are associated to the minimal surfaces, whereas the H2-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 H1-surface/minimal surface classes or in H2-surface/Laguerre minimal surface classes. We also characterize the rotational cases, what allow us finding a complete rotational Laguerre minimal surface.