In this talk we will explore some connections between the analytic notion of pseudoconvexity and the geometric notion of parabolicity. From these connections some older theorems can be proved more simply, and new theorems relating the topology of a Ka Ìhler manifold-with-boundary with the topology of its boundary can be found. Theorems relating the pseudoconvexity of boundary components of Hermitian manifolds to other aspects of its topology have existed since the 60âs. The proofs are analytical, and probably produce limited insight for the geometer. Techniques were developed in the 80âs and 90âs to study the structure of harmonic functions on complete manifolds, and a proof emerged that ALE (asymptotically locally Euclidean) Ka Ìhler manifolds are single-ended. Despite the partial overlap in the conclusions, the methods used appear completely different. Connecting them allows us to produce some new results.