Harmonic mappings and minimal submanifolds

On graphic Bernstein type results in higher codimension

“…In this note, we study the higher codimension case, i.e., minimal submanifolds in R n+m that can be written as graphs of vector-valued functions f : R n → R m . For higher codimension Bernstein type problems, there are general results of [3], [5] and [7]. The idea in these papers is to find a subharmonic function whose vanishing implies that Σ is totally geodesic.…”

“…A fundamental fact is that the Gauss map of a minimal submanifold is a harmonic map; so any convex function on the Grassmannian renders a subharmonic function. The work in [3], [5], and [7] consists of delicate analysis of the geometry of the Grassmannian in order to locate the maximal region where a convex function exists. The condition for Bernstein type results is in terms of the function…”

“…In particular, in [3] and [5], it was shown that * Ω 1 ≥ K > cos p π 2 √ 2p where p = min(n, m) implies that Σ is an affine subspace. This bound is later improved in [7] to…”

Bernstein Type Theorems for Minimal Lagrangian Graphs of Quaternion Euclidean Space

“…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 Jost-Xin [5] established a Bernstein result for M under the condition * Ω ≥ K > 1/2, which improves the previous results in [3] and [2]. Wang in [10] derived a nice Bochner type formula for the function ln( * Ω) −1 .…”

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

“…In [8] Hildebrandt, Jost and Widman have studied entire solutions of the minimal surface system ∂ ∂x i √ gg ij ∂ψ α ∂x j = 0, α = 1, . .…”