Remarks on a Bernstein Type Theorem for Entire Willmore Graphs in R ³

Abstract

In this note we prove that every two-dimensional entire Willmore graph in R ³ with square integrable mean curvature is a plane.

Appendix

Let Σ={(x,y,u(x,y))|(x,y)∈R ²} be a Willmore graph in R ³, where u:R ²→R. Standard calculations then yield that

where $v=\sqrt{1+|Du|^{2}}$. From the calculations in [3] we then get that the Willmore equation (1.2) can be rewritten as follows:

(3.1)

In the paper of [1], the graph condition is used to derive the fact that the image of the Gauss map is contained in the upper hemisphere, which is contractible, and to exclude the case that the density of the entire Willmore graph at infinity is 2. If the graph function depends only on one variable, they ([1], Appendix) proved the theorem without the finiteness condition on the integral of the square of the second fundamental form, by a simple calculation toward the above graphical equation, but in the general case, the graphical equation is not used. Hence, it is natural to guess that there should be a different proof of this theorem that uses the graphical equation essentially, possibly with weaker conditions.