On false branch points of incompressible branched immersions (Q914160)

Branched minimal immersions appear naturally as solutions of Plateau's problem. A basic question is to decide when such an immersion is in fact unbranched. If the minimal surface is area minimizing in a three- manifold, we have a satisfactory answer from the results of Osserman, Alt and Gulliver. In this paper the authors obtain a regularity result under assumptions related to incompressibility of the surface. More precisely, given a branched minimal immersion f from a Riemann surface with boundary M into a Riemannian manifold N, if f induces an isomorphism between the fundamental groups of M and N, and is one-to-one on the boundary of M, then f has no ramified points.