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

Abstract

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space H n is studied. §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 conditions, see [1] and the references therein for codimension-one case). For higher codimensions, the situation becomes more complicated. 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. Warren and Yuan in [9] got a Hession estimate for potential function u : R n → R of minimal Lagrangian graph (x, ∇u) under suitable "convexity" condition, then using the "standard" blow-down process from the geometric measure theory, they got that the potential function u is a quadric polynomial, i.e., the graph (x, ∇u) is an affine plane.In this note, we will investigate a real minimal Lagrangian graph Σ in quaternion space H n = R 4n , which is given by the potential function u : R n → R as follows: Σ = (x, ∇u, ∇u, ∇u). We can use the Bernstein-Pogorelov-Korevaar technique to obtain a Hessian estimate as Warren