A Bernstein Type Result for Special Lagrangian Submanifolds

The well-known Bernstein theorem states any complete minimal surface that can be written as the graph of a function on R must be a plane. This type of result has been generalized in higher dimension and codimension under various conditions. See [2] and the reference therein for the codimension one case and [1], [3], and [6] for higher codimension case. In this note, we prove a Bernstein type result for complete minimal Lagrangian submanifolds of C . We remark that Jost-Xin [7] obtained similar results from a somewhat different approach. Recall a submanifold Σ of C is called Lagrangian if the Kähler form ∑n i=1 dx i ∧ dy restricts to zero on Σ. If Σ happens to be the graph of a vector-valued function from a Lagrangian subspace L to its complement L in C. Rotating C by a element in U(n), we may assume L is the x subspace and L is the y subspace. In this case, there exists a smooth