Illinois Journal of Mathematics

Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$

Abstract

We obtain height estimates for compact embedded surfaces with positive constant mean curvature in a Riemannian product space $\mathbb{M}^{2}\times\mathbb{R}$ and boundary on a slice. We prove that these estimates are optimal for the homogeneous spaces $ℝ^3$, $\mathbb{S}^{2}\times\mathbb{R}$, and $ℍ^{2}×ℝ$ and we characterize the surfaces for which these bounds are achieved. We also give some geometric properties on properly embedded surfaces without boundary.

Article information

Source
Illinois J. Math., Volume 52, Number 1 (2008), 203-211.

Dates
First available in Project Euclid: 15 May 2009

https://projecteuclid.org/euclid.ijm/1242414128

Digital Object Identifier
doi:10.1215/ijm/1242414128

Mathematical Reviews number (MathSciNet)
MR2507241

Zentralblatt MATH identifier
1166.53039

Citation

Aledo, Juan A.; Espinar, José M.; Gálvez, José A. Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$. Illinois J. Math. 52 (2008), no. 1, 203--211. doi:10.1215/ijm/1242414128. https://projecteuclid.org/euclid.ijm/1242414128

References

• U. Abresch and H. Rosenberg, 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.
• X. Cheng and H. Rosenberg, Embedded positive constant $r$-mean curvature hypersurfaces in $\mathbb{M}^m\times\mathbb{R}$, An. Acad. Brasil. Cienc. 72 (2005), 183–199.
• B. Daniel, Isometric immersions into $\mathbb{S} ^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ and applications to minimal surfaces, to appear in T. Amer. Math. Soc.
• I. Fernández and P. Mira, A characterization of constant mean curvature surfaces in homogeneous 3-manifolds, Diff. Geom. Appl. 25 (2007), 281–289.
• J. A. Gálvez and A. Martínez, Estimates in surfaces with positive constant Gauss Curvature, P. Am. Math. Soc. 128 (2000), 3655–3660.
• J. A. Gálvez, A. Martínez and F. Milán, Linear Weingarten surfaces in $\mathbb{R}^3$, Monatsh. Math. 138 (2003), 133–144.
• E. Heinz, On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary, Arch. Rational Mech. Anal. 35 (1969), 249–252.
• D. Hoffman, J. H. S. de Lira and H. Rosenberg, Constant mean curvature surfaces in $\mathbb{M}^2\times\mathbb{R}$, Trans. Amer Math. Soc. 358 (2006), 491–507.
• N. Korevaar, R. Kusner, W. Meeks and B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), 1–43.
• N. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differ. Geom. 30 (1989), 465–503.
• B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.
• H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), 211–239.
• H. Rosenberg and R. Sa Earp, The geometry of properly embedded special surfaces in $\mathbb{R}^3$; e.g., surfaces satisfying $a H +b K =1$, where $a$ and $b$ are positive, Duke Math. J. 73 (1994), 291–306.