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 ̈hler 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 60’s. The proofs are analytical, and probably produce limited insight for the geometer. Techniques were developed in the 80’s and 90’s to study the structure of harmonic functions on complete manifolds, and a proof emerged that ALE (asymptotically locally Euclidean) Ka ̈hler 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.