An upper bound on distance degenerate handle additions

Let V ∪_S W be a Heegaard splitting with a boundary component F. If r is an essential simple closed curve or a slope in F, then there is a Heegaard splitting V (r) ∪_S W obtained by attaching a 2-handle along r on V. It was conjectured by Ma and Qiu that for almost all choices of r, the Heegaard distance d(V (r), W) is the same to d(V, W). By studying handle additions and local properties of the curve complex, we prove that if the distance of V ∪_S W is at least 3, then there is a finite diameter ball in the curve complex C(F) so that it contains all distance degenerate curves or slopes in F. Together with a result proved by Lustig and Moriah, it gives an affirmative answer to Ma and Qiu’ conjecture.