Bernstein Type Theorems for Minimal Lagrangian Graphs of Quaternion Euclidean Space

Abstract

Abstract. In this paper, we prove some Bernstein type results for ndimensional minimal Lagrangian graphs in quaternion Euclidean space H n ∼ = R 4n . In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in C n . §1. IntroductionThe celebrated theorem of Bernstein says that the only entire minimal graphs in Euclidean 3-space are planes. This result has been generalized to R n+1 , for n ≤ 7 and general dimension under various growth condition, see [1] and the reference therein for codimension one case. For higher codimension, the situation becomes more complicated. 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. Let M be a minimal submanifold of R n+m that can be represented as the graph of a smooth map f : R n → R m . The function is given by * Ω = 1 det(I + (df ) t df )

Bernstein Type Results for Lagrangian Graphs with Partially Harmonic Gauss Map

“…The Lagrangian graph in C n has the form of (x, ∇u(x)), where u(x) is a smooth function on R n . Some authors established Bernstein type results for minimal Lagrangian graph under various conditions ( [8], [15], [12], [3]). On the other hand, there are two nice generalizations of minimal Lagrangian submanifolds: Hamiltonian minimal Lagrangian submanifolds and Lagrangian submanifolds with conformal Maslov form.…”

“…Theorem 1 was obtained in [15] under the condition that Σ is minimal Lagrangian and there is some kind of generalization of Yuan's Theorem in [3]; Theorem 2 was established in [6] for the submanifolds of parallel mean curvature in Euclidean space R N and improved in [7] later.…”

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H n

“…Due to the counterexample of Lawson-Osserman [2] , the higher codimension Bernstein type result is not expected to be true in most generality. Hence we have to consider the additional suitable conditions to establish a Bernstein type result of higher codimension.In recent years, remarkable progress has been made by [3][4][5][6][7][8][9] 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.…”

“…Due to string theory, special Lagrangian submanifolds received much attain in recent years. Some authors also tried to establish Bernstein type results for special Lagrangian submanifolds(see [3,[5][6][8][9]). It is known that a special Lagrangian graph may be represented by a gradient of smooth function.…”

“…In recent years, remarkable progress has been made by [3][4][5][6][7][8][9] 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.…”