Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space
type: Item
instance of: scholarly article
title: Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space
zbMATH Open document ID: 1430.53060
author: Luis J. Alías
author: Josué Meléndez
published in: Geometriae Dedicata
publication date: 2019-03-29T00:00:00Z
review text: It has been known by \textit{T. Klotz} and \textit{R. Osserman} [Comment. Math. Helv. 41, 313--318 (1966; Zbl 0163.16002)] that totally umbilical spheres and circular cylinders are the only complete surfaces immersed in 3-dimensional Euclidean space \(\mathbb{R}^3\) with nonzero constant mean curvature and its Gaussian curvature does not change the sign. The higher-dimensional case was investigated by \textit{P. Hartman} [Trans. Am. Math. Soc. 245, 363--374 (1978; Zbl 0412.53027)]. In this paper, the authors generalize the above theorem to the case of hypersurfaces with constant higher-order mean curvatures. Theorem: Let \(n\geq 3\) and \(2\leq r < n\) and let \(M\) be a complete hypersurface immersed into the Euclidean space \(\mathbb{R}^{n+1}\) with constant \(r\)-th mean curvature \(H_{r}\neq 0\) and two distinct principal curvatures, one of them being simple. If its Gauss-Kronecker curvature \(K\) does not change sign, then \(K=0\) and \(M\) is isometric to a cylinder \(\mathbb{R} \times \mathbb{S}^{n+1}(\rho) \subset \mathbb{R}^{n+1}\) with radius \(\rho >0\). As a result, they obtain the same as obtained in [the first author and \textit{S. C. García-Martínez}, Geom. Dedicata 156, 31--47 (2012; Zbl 1232.53046)] for the case \(r=2\).
reviewed by: Yun Myung Oh
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 7045071
zbMATH Keywords: hypersurface
zbMATH Keywords: higher-order mean curvature
zbMATH Keywords: Gauss-Kronecker curvature
zbMATH Keywords: principal curvature theorem
MaRDI profile type: Publication
full work available at URL: s10711-018-0348-z
OpenAlex ID: W2792354554
cites work: An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms
cites work: Hypersurfaces with constant higher order mean curvature in Euclidean space
cites work: Geometry of Hypersurfaces
cites work: Rotation Hypersurfaces in Spaces of Constant Curvature
cites work: On Complete Hypersurfaces of Nonnegative Sectional Curvatures and Constant mth Mean Curvatures
cites work: Complete surfaces in \(E^ 3\) with constant mean curvature
cites work: Q5767686
cites work: On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces
cites work: Minimal Hypersurfaces in a Riemannian Manifold of Constant Curvature
cites work: Proof of the soul conjecture of Cheeger and Gromoll
cites work: Q5770505
cites work: Efimov's theorem in dimension greater than two
cites work: Q3083040
Wikidata QID: Q130069528
DOI: 10.1007/S10711-018-0348-Z
Recommended article: Hypersurfaces with constant higher order mean curvature in Euclidean space
Recommended article: Q3150742
Recommended article: Hypersurfaces with constant higher order mean curvature in space forms
Recommended article: CONSTANT HIGHER-ORDER MEAN CURVATURE HYPERSURFACES IN RIEMANNIAN SPACES
Recommended article: Remarks on hypersurfaces with constant mean curvature in a higher dimensional pseudo-sphere
Recommended article: Hypersurfaces of constant higher-order mean curvature in \(M\times{\mathbb{R}}\)
Recommended article: Q4233031
Recommended article: On hypersurfaces with constant mean curvature in hyperbolic space
Recommended article: A remark on compact hypersurfaces with constant mean curvature in space forms
Recommended article: Compact hypersurfaces with constant higher order mean curvatures
description: scientific article; zbMATH DE number 7045071