Bilateral self-amalgamation of a Heegaard splitting and Hempel distance

A closed orientable Haken three manifold containing a non separating incompressible closed surface has two canonical Heegaard splittings, which are called self-amalgamation and bilateral self-amalgamation. Heegaard distance introduced by Hempel is a useful index in studying Heegaard splitting. This paper studies the stabilization problem for the bilateral self-amalgamation, and proves that if the distance of bilateral selfamalgamation of a Heegaard splitting is at least 9, then it is unstabilized, weakly reducible and irreducible.