Uniform 1-cochains and genuine laminations

Abstract

We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting a transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are the straightening of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when we collapse these laminations, we get a topological pseudo-Anosov flow, as defined by Mosher.

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

“…Then, if Λ induces a π 1 -invariant family of circles in S 2 ∞ ( M ) (up to the continuous extension property [Fe], [Ca]), then by spanning one of the curves in the family with a least area plane Σ, and considering images of Σ under deck transformations, we get a π 1 -invariant family of least area planes in S 2 ∞ ( M ). By the above conjecture, this family is pairwise disjoint; hence it can be shown that it projects down to a embedded genuine lamination with least area leaves in M by using the techniques in [Co1]. Similar construction works for incompressible surfaces, too, which implies a similar result of [HS].…”

“…As described in Remark 4.1, this conjecture has many important applications in 3-manifold topology by combining it with the techniques developed in [Co1]. As Theorem 4.1 shows, to finish this conjecture the only case that needs to be ruled out is when Γ 1 , Γ 2 are not crossing each other and they both bound more than one least area plane, i.e.…”

“…In other words, the following Disjoint Planes Conjecture [Co1] is true if one of the curves bounds a unique least area plane. Remark 4.1.…”

“…Now, we will prove that if one of the curves in the conjecture bounds a unique least area plane, then the conjecture is true. Now, by using the techniques in [Co1], as described in Remark 4.1, for a genuine lamination Λ in a 3-manifold M , with the corollary above, it is generically possible to modify it without changing the metric of M so that the leaves of the new lamination Λ are minimal. Similar, construction works for incompressible surfaces in Gromov hyperbolic 3-manifolds, too.…”

“…∂ ∞ Σ = Γ. The author gave a positive answer to this question and showed the existence of such least area planes asymptotic to a given curve in S 2 ∞ (X) in [Co1]. For the analogous problem for H 3 , Michael Anderson showed the existence of a least area plane Σ ⊂ H 3 asymptotic to a given simple closed curve Γ ⊂ S 2 ∞ (H 3 ) in [A1], [A2].…”