Least Area Planes in Gromov Hyperbolic 3-Spaces with Co-compact Metric

Abstract

Let M be a closed irreducible Riemannian 3-manifold such that π 1 (M) is word hyperbolic, and p : X → M the universal covering. Suppose that X has the Riemannian metric induced form that on M via p. In this paper, we will show that any Jordan curve in the boundary ∂X of X spans a properly embedded least area plane in X. Mathematics Subject Classifications (2000). Primary 57M50; secondary 53A10.

Properly embedded least area planes in Gromov hyperbolic $3$-spaces

“…Then, he generalized his result to Gromov hyperbolic 3-spaces with cocompact metric in [So2]. In this paper, we will show the same results with a simpler and more general technique which is applicable to many different settings.…”

“…Teruhiko Soma gave a positive answer to the Gabai's conjecture in hyperbolic 3-space with cocompact metric in [So1]. Then, he generalized his result to Gromov hyperbolic 3-spaces with cocompact metric in [So2]. In this paper, we will show the same results with a simpler and more general technique which is applicable to many different settings.…”

Number of least area planes in Gromov hyperbolic $3$-spaces

“…Note that the unique planes in the main theorem are all properly embedded by the existence results on properly embedded least area planes in Gromov hyperbolic spaces [So1], [So2] and [Co3]. Note also that by the construction, [Co3] implies that all the canonical planes given by Lemma 3.2 are also proper.…”

Asymptotic H–Plateau problem in ℍ3

“…Gabai conjectures the existence of a properly embedded area minimizing plane in H 3 (and for any cocompact metric on H 3 ) for any given simple closed curve Γ in S 2 ∞ (H 3 ). Later, Soma proved the existence of such an area minimizing plane in more general situation (Gromov hyperbolic spaces) in [50] and [51]. Later, the author gave an alternative proof for Soma's results in [16].…”

“…Theorem 7.1. [50], [51], [16] Let X be a Gromov hyperbolic 3-space with cocompact metric, and S 2 ∞ (H 3 ) be the sphere at infinity of X. Let Γ be a given simple closed curve in S 2 ∞ (H 3 ).…”