Finiteness of the ramified set for branched immersions of surfaces

We shall be concerned with the behavior of a mapping π from one oriented compact surface-with-boundary to another, which may fail to be a covering projection in one of two ways. Firstly, π need not be a local homeomorphism, although its interior singularities will be of a restricted type, called branch points. Secondly, boundary points may be mapped into the interior, although we shall assume the restriction of π to the boundary is injective. We shall show that π must then be a local homeomorphism except on a finite set. Moreover, we shall analyze the behavior of π near the boundary in sufficient detail to derive a formula relating Euler characteristics of the domain and of the image, with multiplicities, to the total order of branching of π. These results may be used to study ramification and ramified branch points of parametric minimal surfaces of general topological type.