The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic -manifold has at most finitely many irreducible Heegaard splittings. Along the way we show that has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index- minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic -manifolds.
"Effective finiteness of irreducible Heegaard splittings of non-Haken -manifolds." Duke Math. J. 167 (15) 2793 - 2832, 15 October 2018. https://doi.org/10.1215/00127094-2018-0022