Bernstein type theorems for higher codimension

Abstract

We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds.

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…”

“…Notice that K is not necessarily bounded from below. At the end of §3, we discuss how Theorem A implies the results in [3], [5] or [7].…”

Mean curvature flow via convex functions on Grassmannian manifolds

“…This was found in a previous work of Jost-Xin [6]. For any real number a let V a = {P ∈ G n,m , v(P ) < a}.…”