Finiteness of Disjoint Minimal Graphs

of u in R is called a minimal graph supported on Ω. In a recent article of Meeks-Rosenberg [M-R], where they proved the unicity of the helicoid, the authors showed that if the defining functions {ui} of a set of disjointly supported minimal graphs {Gi} have bounded gradients, then the number of graphs must be finite. In a private communication with the first author, Rosenberg posed the question if the number of disjoint minimal graphs, whose defining functions are at most polynomial growth of a fixed degree, is finite. Obviously, this question was motivated by his work with Meeks, but it was also related to the type of finiteness theorems the first author proved in [L] concerning harmonic functions. This argument was later generalized by the authors [L-W] to show the finiteness of disjoint d-massive sets and was used to prove a structural theorem for harmonic maps. It turns out that this argument can also be used to study disjoint minimal graphs. Indeed, the purpose of this note is to show that there are only finitely many minimal graphs supported on disjoint open subsets in R. Moreover, we will prove that the maximum possible number of such disjointly supported minimal graphs is (n + 1)2. We would like to point out that it is somewhat surprising that