Effective finiteness of irreducible Heegaard splittings of non-Haken 3-manifolds

Abstract

The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic $3$-manifold $N$ has at most finitely many irreducible Heegaard splittings. Along the way we show that $N$ has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index-$\le 1$ minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic $3$-manifolds.