Effective hyperbolization and length bounds for Heegaard splittings

TL;DR

This paper establishes conditions under which a 3-manifold with a Heegaard splitting admits a hyperbolic metric and provides bounds on the length of certain curves based on topological data, without relying on Geometrization.

Contribution

It introduces topological criteria for hyperbolicity of 3-manifolds from Heegaard splittings and derives explicit length bounds for curves in terms of disk set projections.

Findings

Manifolds with sufficiently incompressible curves are hyperbolic.

Length of curves can be estimated from disk set projection coefficients.

Hyperbolicity can be deduced without Geometrization theorem.

Abstract

We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_\Sigma H^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular, given a suitable "sufficiently incompressible" curve $\gamma\subset\Sigma$, we show (without appealing to Geometrization) that $M$ is hyperbolic and we compute the length of $\gamma$ in terms of the projection coefficients of the disk sets, up to a uniform multiplicative error.