Level Curves of Minimal Graphs

Abstract

We consider minimal graphs u = u(x, y) > 0 over domains D ⊂ R 2 bounded by an unbounded Jordan arc γ on which u = 0. We prove an inequality on the curvature of the level curves of u, and prove that if D is concave, then the sets u(x, y) > C (C > 0) are all concave. A consequence of this is that solutions, in the case where D is concave, are also superharmonic.