Minimal Heegaard Genera of 3-manifolds with Lower Distance

For a 3-manifold M, the genus of M, denoted by g(M), is defined to be the minimal Heegaard genus among all the Heegaard splittings of M. In this paper, we prove that for any two integers g ≥ 2 and n ≥ 2, there is a 3-manifold M with g(M) = g such that the minimal Heegaard splittings of M are unique up to isotopy, where the distance of the Heegaard splitting is n.