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

Suppose a closed orientable 3-manifold M has a genus g Heegaard surface P with distance d ( P ) = 2 g . 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 2 g + 2 critical points p 0 , p 1 , … , p 2 g + 1 with f ( p 0 ) < f ( p 1 ) < ⋯ < f ( p 2 g + 1 ) where p 0 is a minimum and p 2 g + 1 is a maximum, and p 1 , … , p 2 g are saddles, (2) if we take regular values r i ( i = 1 , … , 2 g + 1 ) such that f ( p i − 1 ) < r i < f ( p i ) , then f − 1 ( r i ) ∩ Q consists of a circle if i is odd, and f − 1 ( r i ) ∩ Q consists of two circles if i is even.