Isotoping Heegaard surfaces in neat positions with respect to critical distance Heegaard surfaces

AbstractSuppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d(P)=2g. Let Q be another genus g Heegaard surface which is strongly irreducible. Then we show that there is a height function f:M→I induced from P such that by isotopy, Q is deformed into a position satisfying the following; (1) f|Q has 2g+2 critical points p0,p1,…,p2g+1 with f(p0)<f(p1)<⋯<f(p2g+1) where p0 is a minimum and p2g+1 is a maximum, and p1,…,p2g are saddles, (2) if we take regular values ri (i=1,…,2g+1) such that f(pi−1)<ri