On the existence of minimal Heegaard surfaces

Abstract

Let H be a strongly irreducible Heegaard surface in a closed oriented Riemannian 3-manifold. We prove that H is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein. In the case of positive scalar curvature, we show for spherical space forms not diffeomorphic to S³ or RP³ that any strongly irreducible Heegaard splitting is isotopic to a minimal surface, and that there is a minimal Heegaard splitting of area less than $4π$ if R ≥ 6.