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Translating solitons of the mean curvature flow in the space \({\mathbb {H}}^2\times {\mathbb {R}}\)

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Abstract

In this paper we study the theory of translating solitons of the mean curvature flow of immersed surfaces in the product space \({\mathbb {H}}^2\times {\mathbb {R}}\). We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a conformal metric space. Explicit examples of these surfaces are constructed, and we study the asymptotic behavior of the existing rotationally symmetric examples. Finally, we prove some uniqueness and non-existence theorems.

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Acknowledgements

The author is grateful to the referee for helpful comments that highly improved the final version of the paper.

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Authors and Affiliations

  1. Departamento de Geometráa y Topología, Universidad de Granada, 18071, Granada, Spain

    Antonio Bueno

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  1. Antonio Bueno
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Correspondence to Antonio Bueno.

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The author was partially supported by MICINN-FEDER, Grant No. MTM2016-80313-P and Junta de Andalucía Grant No. FQM325.

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Bueno, A. Translating solitons of the mean curvature flow in the space \({\mathbb {H}}^2\times {\mathbb {R}}\). J. Geom. 109, 42 (2018). https://doi.org/10.1007/s00022-018-0447-x

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  • DOI: https://doi.org/10.1007/s00022-018-0447-x

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