When is a minimal surface a minimal graph? (Q1879994)

The paper studies the conformal structures of non-compact, proper, branched minimal surfaces in \(\mathbb R^3\) and proves several criteria for such surfaces to be parabolic. Two main theorems are stated and proved. On short, theorem 1.1 extends Rado's theorem to the case when the projection is no longer compact. Theorem 1.2 gives a necessary and sufficient condition for the existence of a certain (proper, branched) minimal surface whose interior is a Jenkins-Serrin graph over a planar domain bounded by a convex \(n\)-gon. Theorem 1.2 may be viewed as an extension of a result given by Langevin, Levitt and Rosenberg in the late 1980s.