When is a Minimal Surface a Minimal Graph

Following Parreau's work in 1951-52, we give a unified definition of parabolic Riemann surfaces, with or without boundary. A surface is parabolic under the unified definition implies that it is either relative parabolic or parabolic under the classical definitions. Then we study the conformal structures of noncompact, proper, branched minimal surfaces in R 3 and prove several criteria of such surfaces (with or without boundary) being parabolic. Using these criteria we then prove two graph theorems, they are noncompact versions of the classical graph theorem of Rado, generalized in various directions.