On boundary primitive manifolds and a theorem of Casson–Gordon

In this paper it is shown that manifolds admitting minimal genus weakly reducible but irreducible Heegaard splittings contain an essential surface. This is an extension of a well-known theorem of Casson–Gordon to manifolds with non-empty boundary. The situation for non-minimal genus Heegaard splittings is also investigated and it is shown that boundary stabilizations are stabilizations for manifolds which are boundary primitive.