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The slab theorem for minimal surfaces in \(\mathbb {E}(-1,\tau )\)

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Abstract

Unlike \(\mathbb {R}^{3}\), the homogeneous spaces \(\mathbb {E}(-1,\tau )\) have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in \(\mathbb {E}(-1,\tau )\) is such a graph. More specifically, we introduce the definition of a generalized slab in \(\mathbb {E}(-1,\tau )\) and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.

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Acknowledgments

This work is part of the author’s Ph.D. thesis at Instituto Nacional de Matemática Pra e Aplicada (IMPA). The author would like to express his sincere gratitude to his advisor Harold Rosenberg for his patience, constant encouragement and guidance. He would like also to thank Marco A. M. Guaraco for making the pictures that appear in the paper, Abigail Folha and Carlos Peñafiel for provide the preprint [9], and Benoît Daniel, José Espinar, Laurent Hauswirth, Laurent Mazet, Lúcio Rodriguez, Magdalena Rodriguez and William Meeks III for discussions and their interesting in this work. Finally he also thanks the referee by the suggestions and corrections.

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  1. Instituto de Matemática e Estatística, UERJ Rua São Francisco Xavier, 524 Pavilhão Reitor João Lyra Filho, 6 andar-Bloco B, 20550-900, Rio de Janeiro, RJ, Brazil

    Vanderson Lima

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  1. Vanderson Lima
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Correspondence to Vanderson Lima.

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The author was supported by CNPq-Brazil.

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Lima, V. The slab theorem for minimal surfaces in \(\mathbb {E}(-1,\tau )\) . Ann Glob Anal Geom 51, 189–208 (2017). https://doi.org/10.1007/s10455-016-9531-3

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  • DOI: https://doi.org/10.1007/s10455-016-9531-3

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