Abstract
We prove that a H-surface M in \({\mathbb{H}}^2 \times {\mathbb{R}} ,\vert H\vert \leq 1/2\) , inherits the symmetries of its boundary \(\partial M,\) when \(\partial M\) is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.
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The second and the third authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.
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Nelli, B., Sa Earp, R., Santos, W. et al. Uniqueness of H-surfaces in \({\mathbb{H}}^2 \times \mathbb{R},{{\vert H\vert \leq 1/2}}\) , with boundary one or two parallel horizontal circles. Ann Glob Anal Geom 33, 307–321 (2008). https://doi.org/10.1007/s10455-007-9087-3
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DOI: https://doi.org/10.1007/s10455-007-9087-3
Keywords
- Product space \({\mathbb{H}}^2 \times \mathbb {R}\)
- Mean curvature \(\leq\frac{1}{2}\)
- Boundary a circle
- Symmetries
- Abresch–Rosenberg holomorphic quadratic differential
- Vertical graph
- Rotational H-annulus
- Asymptotic boundary