Effective hyperbolization and length bounds for Heegaard splittings

We consider 3-manifolds given as Heegaard splittings M=H⁻∪_Σ 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 γ⊂Σ, we show (without appealing to Geometrization) that M is hyperbolic and we compute the length of γ in terms of the projection coefficients of the disk sets, up to a uniform multiplicative error.