# sequence of graphs converge in the sense of varifold to multiplicity 2 plane

Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) to a limit V, which is multiplicity 2 of the plane P? Seems impossible but is there a proof?

• It's possible to converge to a multiplicity 2 plane which is perpendicular to P though. – Sam Jun 13 '16 at 20:15

I think that the following argument rules out getting $2[P]$:
Suppose that $v_n : = graph (f_n) \rightharpoonup 2[P]$ as varifolds. Consider the open set $U = \{(x,P') \in \mathbb{R}^3\times Gr(2,3) : |x| < 1, d(P,P') <\epsilon\}$. Then, a standard result for weak convergence of measures implies that $$\liminf_{n\to\infty} v_n(U) \geq 2[P](U) \qquad (= 2 |B_1(0)|).$$ However, it is easy to bound $v_n(U) \leq |B_1(0)| + O(\epsilon)$ as $\epsilon\to0$, since $v_n(U)$ is the $\mathcal{H}^2$ measure of the set of points in $graph(f_n) \cap B_1(0)$ whose normal vectors differ from that of $P$ by a distance $O(\epsilon)$ (the precise distance we choose on $Gr(2,3)$ is irrelevant). This is a contradiction.
• But the $\mathcal{H}^2$ measure of the graph is getting close to $2|B_1(0)|$. And more and more of it could have normal vector far from the normal to $P$. For example, isn't it fairly clear you can converge to $\sqrt{2}[P]$ by zig-zagging? – Thompson Oct 20 '16 at 18:17