Heegaard genera of high distance are additive under annulus sum

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on ∂Mi, i=1,2. Let h:A1→A2 be a homeomorphism, and M=M1∪hM2 the annulus sum of M1 and M2 along A1 and A2. In the present paper, we show that if Mi has a Heegaard splitting Vi∪SiWi with distance d(Si)⩾2g(Mi)+3 for i=1,2, then g(M)=g(M1)+g(M2). Moreover, if g(Fi)⩾2, i=1,2, then the minimal Heegaard splitting of M is unique.