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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Embedded $ H$-planes in hyperbolic $ 3$-space


Author: Baris Coskunuzer
Journal: Trans. Amer. Math. Soc. 371 (2019), 1253-1269
MSC (2010): Primary 53A10, 57M50; Secondary 53C42
DOI: https://doi.org/10.1090/tran/7286
Published electronically: July 31, 2018
MathSciNet review: 3885178
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Abstract: We show that for any $ \mathcal {C}^0$ Jordan curve $ \Gamma $ in $ S^2_{\infty }(\mathbf {H}^3)$, there exists an embedded $ H$-plane $ \mathcal {P}_H$ in $ \mathbf {H}^3$ with $ \partial _{\infty } \mathcal {P}_H =\Gamma $ for any $ H\in (-1,1)$. As a corollary, we prove that any quasi-Fuchsian hyperbolic $ 3$-manifold $ M\simeq \Sigma \times \mathbb{R}$ contains an $ H$-surface $ \Sigma _H$ in the homotopy class of the core surface $ \Sigma $ for any $ H\in (-1,1)$. We also prove that for any $ C^1$ Jordan curve in $ S^2_{\infty }(\mathbf {H}^3)$, there exists a unique minimizing $ H$-plane $ \mathcal {P}_H$ with $ \partial _{\infty } \mathcal {P}_H =\Gamma $ for a generic $ H\in (-1,1)$.


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Additional Information

Baris Coskunuzer
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: coskunuz@bc.edu

DOI: https://doi.org/10.1090/tran/7286
Received by editor(s): March 20, 2017
Received by editor(s) in revised form: May 8, 2017, and May 19, 2017
Published electronically: July 31, 2018
Additional Notes: The author is partially supported by a BAGEP award of the Science Academy, Simons Collaboration Grant, and Royal Society Newton Mobility Grant
Article copyright: © Copyright 2018 American Mathematical Society