In this paper, branched
immersions between compact orientable 2-manifolds are considered. Branched
immersions are smooth maps whose only singularities are branch points, i.e.,
points of the domain where the map is locally topologically equivalent to
z → zr(r = 2,3,⋯). Originally these maps were studied in connection with Douglas’
solution to Plateau’s problem.
The maps considered here are required to satisfy natural boundary hypothesis
which have been motivated by minimal surface studles. The main result completely
decides the existence question for a branched immersion between compact orientable
2-manifolds with or without boundary.
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