Number of least area planes in Gromov hyperbolic 3-spaces

Coskunuzer, Baris

doi:10.1090/S0002-9939-10-10308-6

Number of least area planes in Gromov hyperbolic $3$-spaces
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Proc. Amer. Math. Soc. 138 (2010), 2923-2937 Request permission

Abstract:

We show that for a generic simple closed curve $\Gamma$ in the asymptotic boundary of a Gromov hyperbolic $3$-space with cocompact metric $X$, there exists a unique least area plane $\Sigma$ in $X$ such that $\partial _{\infty }\Sigma = \Gamma$. This result has interesting topological applications for constructions of canonical $2$-dimensional objects in Gromov hyperbolic $3$-manifolds.

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