A Bernstein problem for special Lagrangian equations

Abstract

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.Comment: 9 pages, submitted on December 10, 200

Bernstein Type Theorems for Minimal Lagrangian Graphs of Quaternion Euclidean Space

“…A useful formula for the minimal Lagrangian graph is derived by applying Wang's Bochner formula to the quaternionic frame. Using this formula, we obtain some Bernstein theorems for Σ, which generalize those results in [8] and [11] (see Section 3 for details). Obviously, when u 2 and u 3 are constant, Σ is just the special Lagrangian graph in C n .…”

“…Tsui and Wang in [8] obtained Bernstein results for special Lagrangian graphs by applying the same Bochner formula. We should point out that the same formula for special Lagrangian graphs was also derived by Yuan in [11] from a different point of view. An important technique used by Yuan is the so-called Lewy transformation which allows him to prove: Any special Lagrangian graph given by a convex potential function must be an affine plane.…”

“…Some authors also tried to establish Bernstein type results for special Lagrangian submanifolds (see [6], [8] and [11]). It is known that a special Lagrangian graph may be represented by a gradient of a smooth function, i.e., f = ∇u for a smooth function u : R n → R. The function u is called the potential function.…”

“…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.…”

Entire mean curvature flows of graphs

“…The advantage of this kind of map is that we can find a good representation of via Lewy transformation. This technique is used by Yuan [2002] to prove a Bernstein theorem for special Lagrangian graphs. The difficult is that this case is noncompact.…”