Hypersurfaces with radial mean curvature in space forms

Carlos Maber Carrión Riveros,Armando M Vasquez Corro,Edwin O Salinas Reyes

doi:10.5216/nm.v4.70251

Abstract

In this paper, we study two classes of hypersurfaces, namely, the DRMC-hypersurfaces and the HDRMC-hypersurfaces in space forms Mn+1(c), c=−1,0,1, these classes include the Weingarten hypersurfaces of the spherical type obtained in |10|. For n= 2, we present a way to obtain DRMC-surfaces and HDRMC-surfaces in M3(c) using two holomorphic functions. Also, we classify the DRMC-hypersurfaces of rotation in Mn+1(c) and the HDRMC-hypersurfaces of rotation in Rn+1.

Highlights

The surfaces M ⊂ R3 satisfying a functional relation of the form W (H, K) = 0, where H and K are the mean and Gaussian curvatures of the surface M, respectively, are called Weingarten surfaces
Examples of Weingarten surfaces are the surfaces of revolution and the surfaces of constant mean or Gaussian curvature
In [2] presented a way of parameterizing surfaces as envelopes of a congruence of spheres in which an envelope is contained in a plane and with radius function h associated with a hydrodynamic type system. It studies the surfaces in hyperbolic space H3 satisfying the relation

Summary

Introduction

In [2] presented a way of parameterizing surfaces as envelopes of a congruence of spheres in which an envelope is contained in a plane and with radius function h associated with a hydrodynamic type system As an application, it studies the surfaces in hyperbolic space H3 satisfying the relation. In [9], the author present a way to parameterize hypersurfaces as congruence of spheres in which an envelope is contained in a hyperplane Using this parametrization is presented a generalization of the surfaces of the spherical type (Laguerre minimal surfaces) studied in [8], namely the Weingarten hypersurfaces of the spherical type, i.e. the oriented hypersurfaces of the Euclidean space M ⊂ Rn+1 satisfying a Weingarten relation of the form n (−1)r+1rf r−1 n r r=1 where f ∈ C∞(M ; R) and Hr is the rth mean curvature of M. We classify the DRMC-hypersurfaces of rotation in M n+1(c) and the HDRMC-hypersurfaces of rotation in Rn+1

Preliminaries

Hypersurfaces with radial mean curvature

Conclusions