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Concentration of total curvature of minimal surfaces in \(\mathbb {H}^2\times \mathbb R\)

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Abstract

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space \(\mathbb H^2 \times \mathbb R,\) where \(\mathbb H^2\) is the hyperbolic plane. Under some geometric conditions on its asymptotic boundary, an oriented stable minimal surface immersed in \(\mathbb H^2 \times \mathbb R\) has infinite total curvature. We exhibit an example of a minimal graph such that in a domain whose asymptotic boundary is a vertical segment the total curvature is finite, but the total curvature of the graph is infinite, by the theorem cited before. We also present some simple and peculiar examples of infinite total curvature minimal surfaces in \(\mathbb H^2 \times \mathbb R\) and their asymptotic boundaries.

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Acknowledgements

The second author wishes to thank the Departamento de Matemática da PUC-Rio for their kind hospitality. The authors are grateful to the referee for valuable suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, 22451-900, Brazil

    Ricardo Sa Earp

  2. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot-Paris 7, Equipe Géométrie et Dynamique, UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    Eric Toubiana

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  1. Ricardo Sa Earp
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  2. Eric Toubiana
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Correspondence to Eric Toubiana.

Additional information

R. Sa Earp was partially supported by CNPq of Brasil.

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Sa Earp, R., Toubiana, E. Concentration of total curvature of minimal surfaces in \(\mathbb {H}^2\times \mathbb R\) . Math. Ann. 369, 1599–1621 (2017). https://doi.org/10.1007/s00208-016-1508-9

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  • DOI: https://doi.org/10.1007/s00208-016-1508-9

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