Heegaard genera of high distance are additive under annulus sum

Let M i be a compact orientable 3-manifold, and A i a non-separating incompressible annulus on ∂ M i , i = 1 , 2 . Let h : A 1 → A 2 be a homeomorphism, and M = M 1 ∪ h M 2 the annulus sum of M 1 and M 2 along A 1 and A 2 . In the present paper, we show that if M i has a Heegaard splitting V i ∪ S i W i with distance d ( S i ) ⩾ 2 g ( M i ) + 3 for i = 1 , 2 , then g ( M ) = g ( M 1 ) + g ( M 2 ) . Moreover, if g ( F i ) ⩾ 2 , i = 1 , 2 , then the minimal Heegaard splitting of M is unique.