Effective hyperbolization and length bounds for Heegaard splittings

We consider 3-manifolds given as Heegaard splittings $M=H^{-}{\cup }_{\mathrm{\Sigma }}{H}^{+}$ 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 $\gamma \subset \mathrm{\Sigma }$, 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.