A Bernstein Type Result for Special Lagrangian Submanifolds

Abstract

Abstract. Let Σ be a complete minimal Lagrangian submanifold of C n . We identify several regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σ lies in one of these regions, then Σ is an affine space.

Bernstein Type Theorems for Minimal Lagrangian Graphs of Quaternion Euclidean Space

“…If u 2 and u 3 are constants, then Σ = (x, ∇u 1 (x), 0, 0), which may be regarded as a minimal Lagrangian graph Σ = (x, ∇u 1 (x)) in C n . So the above Corollary generalizes those results in [8] and [11].…”

“…Note that if u 2 , u 3 are constants, Σ is just the special Lagrangian graph considered in [6], [8] and [11].…”

“…Due to the counterexample of , the higher codimension Bernstein type result is not expected to be true in the most generality. Hence we have to consider the additional suitable conditions to establish a Bernstein type result for higher codimension.In recent years, remarkable progress has been made by [5], [6], [8], [10] and [11] in Bernstein type problems of minimal submanifolds with higher codimension and special Lagrangian submanifolds. The key idea in these papers is to find a suitable subharmonic function, whose vanishing implies the minimal graph is totally geodesic.…”

On graphic Bernstein type results in higher codimension

“…If we require P + AQ to be invertible, then the condition is determined by the eigenvalues of the symmetric matrix (P + AQ) −1 (−Q + AP ). Please see [11] for the optimal condition in this case.…”

“…These include holomorphic submanifolds, special Lagrangian submanifolds, associative, co-associative, and Cayley submanifolds. The special Lagrangian case is studied in [4], [10], [8], [14], and [11]. The Lagrangian condition implies the existence of a potential function F such that f = ∇F .…”