A Sharp Bound for the Growth of Minimal Graphs

We consider minimal graphs u = u ( x , y ) > 0 over unbounded domains D ⊂ R 2 bounded by a Jordan arc γ on which u = 0 . We prove a sort of reverse Phragmén-Lindelöf theorem by showing that if D contains a sector S λ = { ( r , θ ) = { - λ / 2 < θ < λ / 2 } , π < λ ≤ 2 π , then the rate of growth is at most r π / λ .