Abstract
Let \(\fancyscript{F}(N\times \mathbb{R})\) be the set of all closed H-hypersurfaces \(M\subset N\times \mathbb{R}\) , where N is a simply connected complete Riemannian n-manifold with sectional curvature K N ≤ −κ2 < 0. We show that \(\rule{.5pt}{6.8pt}{\kern-.6pt}{\rm h}(N \times \mathbb{R})=\inf_{M\in{\fancyscript{F}}(N\times \mathbb{R})}\{\vert H_{M}\vert \}\geq (n-1)\kappa/n\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abresech, U., Rosenberg, H. (2004) 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(2):141–174
Barbosa J.L.M., Kenmotsu K. and Oshikiri O. (1991). Foliations by hypersurfaces with constant mean curvature. Math. Z. 207: 97–108
Bessa G.P. and Montenegro J.F. (2003). Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24: 279–290
Bessa G.P., Jorge L., Lima B. and Montenegro J.F. (2006). Fundamental tone estimates for elliptic operators in divergence form and geometric applications Math. DG/0403436. An. Acad. Bras. Ciênc. 78: 391–404
Choe J. and Gulliver R. (1992). Isoperimetric inequalities on minimal submanifolds of space forms. Manuscripta Math. 77: 169–189
Cheung, Leung-Fu, Leung, Pui-Fai. (2001) Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236:525–530
Frankel T. (1966). On the fundamental group of a compact minimal submanifold. Ann. Math. 83: 68–73
Hsiang W.T. and Hsiang W.Y. (1989). On the uniqueness of isoperimetric solutions and embedded soap bubles in non-compact symmetric spaces I. Inv. Math. 98: 39–58
Jorge L. and Koutrofiotis D. (1980). An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103: 711–725
Jorge L. and Xavier F. (1981). An inequality between the exterior diameter and the mean curvature of bounded immersions. Math. Z. 178: 77–82
Nelli, B., Rosenberg, H. Global properties of constant mean curvature surfaces in \({\mathbb{H}}^{2}\times {\mathbb{R}}\) . To appear in Pacific. J. Math. (2005)
Rosenberg, H. Plenary conference in the First Meeting in Differential Geometry of UFRJ Rio de janeiro (2005)
Salavessa, I. Graphs with parallel mean curvature and a variational problem in conformal geometry, Ph.D. Thesis, University of Warick (1988)
Salavessa, I. Graphs with parallel mean curvature. Proc. Am. Math. Soc. 107:449–458 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pacelli Bessa, G., Fabio Montenegro, J. On compact H-hypersurfaces of N × \(\mathbb{R}\) . Geom Dedicata 127, 1–5 (2007). https://doi.org/10.1007/s10711-007-9145-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-007-9145-9