## Area estimates and rigidity of non-compact $H$-surfaces in 3-manifolds

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## Abstract:

For appropriate values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a 3-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighborhood of the mean convex side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also, we provide a counterexample showing that, in the case of minimal surfaces, equality in the area estimate does not necessarily imply a local rigidity result for the ambient manifold.## References

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## Additional Information

**Vanderson Lima**- Affiliation: Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre - RS 90040-060 Brazil
- MR Author ID: 1200287
- ORCID: 0000-0003-3740-2348
- Email: vanderson.lima@ufrgs.br
- Received by editor(s): May 15, 2018
- Received by editor(s) in revised form: February 1, 2019
- Published electronically: May 17, 2019
- Additional Notes: This work started while the author was working at Instituto Nacional de Matemática Pura e Aplicada (IMPA) under the fund Programa de Capacitação Institucional PCI/MCTI-CNPq.
- Communicated by: Jiaping Wang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4499-4512 - MSC (2010): Primary 53A10, 53Axx
- DOI: https://doi.org/10.1090/proc/14578
- MathSciNet review: 4002559