A Note on Surfaces in $\mathbb{H}^2 \times \mathbb{R}$

Stefano Montaldo; Irene I. Onnis

A Note on Surfaces in $\mathbb{H}^{2}\times\mathbb{R}$

Stefano Montaldo; Irene I. Onnis

Bollettino dell'Unione Matematica Italiana (2007)

Volume: 10-B, Issue: 3, page 939-950
ISSN: 0392-4041

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Abstract

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In this article we consider surfaces in the product space

\mathbb{H}^{2}\times\mathbb{R}

of the hyperbolic plane

\mathbb{H}^{2}

with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the local classification of totally umbilical surfaces.

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Montaldo, Stefano, and Onnis, Irene I.. "A Note on Surfaces in $\mathbb{H}^2 \times \mathbb{R}$." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 939-950. <http://eudml.org/doc/290411>.

@article{Montaldo2007,
abstract = {In this article we consider surfaces in the product space $\mathbb\{H\}^2 \times \mathbb\{R\}$ of the hyperbolic plane $\mathbb\{H\}^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the local classification of totally umbilical surfaces.},
author = {Montaldo, Stefano, Onnis, Irene I.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {939-950},
publisher = {Unione Matematica Italiana},
title = {A Note on Surfaces in $\mathbb\{H\}^2 \times \mathbb\{R\}$},
url = {http://eudml.org/doc/290411},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Montaldo, Stefano
AU - Onnis, Irene I.
TI - A Note on Surfaces in $\mathbb{H}^2 \times \mathbb{R}$
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 939
EP - 950
AB - In this article we consider surfaces in the product space $\mathbb{H}^2 \times \mathbb{R}$ of the hyperbolic plane $\mathbb{H}^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete minimal graphs; the local classification of totally umbilical surfaces.
LA - eng
UR - http://eudml.org/doc/290411
ER -

References

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ABRESCH, U. - ROSENBERG, H., A Hopf differential for constant mean curvature surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$ , Acta Math., 193 (2004), 141-174. MR2134864 DOI10.1007/BF02392562
CADDEO, R. - PIU, P. - RATTO, A., SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces, Manuscripta Math., 87 (1995), 1-12. Zbl0827.53009 MR1329436 DOI10.1007/BF02570457
DAJCZER, M., Submanifolds and isometric immersions, Mathematics Lecture Series, 13. Publish or Perish, Houston, 1990. MR1075013
EELLS, J. - SAMPSON, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. Zbl0122.40102 MR164306 DOI10.2307/2373037
FERNANDEZ, I. - MIRA, P., Harmonic maps and constant mean curvature surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ , arXiv:math.DG/0507386. MR2343386 DOI10.1353/ajm.2007.0023
MONTALDO, S. - ONNIS, I. I., Invariant CMC surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ , Glasg. Math. J., 46 (2004), 311-321. Zbl1055.53045 MR2062613 DOI10.1017/S001708950400179X
MONTALDO, S. - ONNIS, I. I., Invariant surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ with constant (Gauss or mean) curvature, Publ. de la RSME, 9 (2005), 91-103.
MEEKS III, W. - ROSENBERG, H., The theory of minimal surfaces in $M\times\mathbb{R}$ , Comment. Math. Helv., 80 (2005), 811-858. Zbl1085.53049 MR2182702 DOI10.4171/CMH/36
NELLI, B. - ROSENBERG, H., Minimal surfaces in $\mathbb{H}^{2}\times R$ , Bull Braz. Math. Soc., 33 (2002), 263-292. MR1940353 DOI10.1007/s005740200013
NELLI, B. - ROSENBERG, H., Global properties of constant mean curvature surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ , Pacific J. Math., to appear. MR2247859 DOI10.2140/pjm.2006.226.137
ONNIS, I. I., Superfícies em certos espaços homogêneos tridimensionais, Ph.D. Thesis, University of Campinas (2005), available online at http://libdigi.unicamp.br/document/?code=vtls000364041.
ONNIS, I. I., Geometria delle superfici in certi spazi omogenei tridimensionali, Boll. Un. Mat. Ital. A (8) 9 (2006), 267-270.
OSSERMAN, R., Minimal surfaces in $\mathbb{R}^{3}$ , Global differential geometry, MAA Stud. Math., 27, Math. Assoc. America, Washington, DC, (1989), 73-98. MR1013809
ROSENBERG, H., Minimal surfaces in $M\times\mathbb{R}$ , Illinois Jour. Math., 46 (2002), 1177-1195. Zbl1036.53008 MR1988257
SAÂ EARP, R. - TOUBIANA, E., Screw motion surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and $\mathbb{S}^{2}\times\mathbb{R}$ , Illinois J. Math., 49 (2005), 1323-1362. Zbl1093.53068 MR2210365

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