Weingarten hypersurfaces of the spherical type in Euclidean spaces

Cid D. F. Machado; Carlos M. C. Riveros

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 2, page 213-236
  • ISSN: 0010-2628

Abstract

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We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces M n , n ≥ 2 , in Euclidean space satisfying a relation ∑ r = 1 n ( - 1 ) r + 1 r f r - 1 n r H r = 0 , where H r is the r th mean curvature and f ∈ C ∞ ( M n ; ℝ ) , these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic applications. Also, we classify the Weingarten hypersurfaces of the spherical type of rotation and we give explicit examples.

How to cite

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Machado, Cid D. F., and Riveros, Carlos M. C.. "Weingarten hypersurfaces of the spherical type in Euclidean spaces." Commentationes Mathematicae Universitatis Carolinae 61.2 (2020): 213-236. <http://eudml.org/doc/297242>.

@article{Machado2020,
abstract = {We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\ge 2$, in Euclidean space satisfying a relation $\sum _\{r=1\}^\{n\} (-1)^\{r+1\}rf^\{r-1\} \{ n \atopwithdelims ()r\}H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^\{\infty \}(M^n;\mathbb \{R\})$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic applications. Also, we classify the Weingarten hypersurfaces of the spherical type of rotation and we give explicit examples.},
author = {Machado, Cid D. F., Riveros, Carlos M. C.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Weingarten hypersurface; Laguerre minimal surface; $r$th mean curvature; Laplace–Beltrami operator},
language = {eng},
number = {2},
pages = {213-236},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weingarten hypersurfaces of the spherical type in Euclidean spaces},
url = {http://eudml.org/doc/297242},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Machado, Cid D. F.
AU - Riveros, Carlos M. C.
TI - Weingarten hypersurfaces of the spherical type in Euclidean spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 2
SP - 213
EP - 236
AB - We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\ge 2$, in Euclidean space satisfying a relation $\sum _{r=1}^{n} (-1)^{r+1}rf^{r-1} { n \atopwithdelims ()r}H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty }(M^n;\mathbb {R})$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic applications. Also, we classify the Weingarten hypersurfaces of the spherical type of rotation and we give explicit examples.
LA - eng
KW - Weingarten hypersurface; Laguerre minimal surface; $r$th mean curvature; Laplace–Beltrami operator
UR - http://eudml.org/doc/297242
ER -

References

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