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Let M be a compact orientable 3-manifold, and A an essential annulus which cuts M into two 3-manifolds M1 and M2. We denote by g(M) the Heegaard genus of M. In this paper, we will give a lower bound to g(M) when Mi contains no bounded essential surfaces with large Euler's characteristic, where all boundary components of the essential surfaces lie in A.

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Let M be a compact orientable irreducible 3-manifold, and F be an essential connected closed surface in M which cuts M into two manifolds M1 and M2. If Mi has a minimal Heegaard splitting Mi = Vi ∪H i Wi with d(H1) + d(H2) ≥ 2(g(M1) + g(M2) − g(F )) + 1, then g(M ) = g(M1) + g(M2) − g(F ).