A growth gap for anisotropic minimal graphs

Abstract

We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each $n\ge 2$ and each smooth uniformly elliptic parametric integrand $\Phi$ on ${\mathbb{R}}^{n+1}$, there is an exponent $\alpha =\alpha (n,\Phi )>0$ such that every smooth nonaffine entire $\Phi$-minimal graph $u:{\mathbb{R}}^n\to \mathbb{R}$ satisfies ${\sup }_{B_R^n}|Du|\ge c{R}^{\alpha }$ for all $R\ge R_0$, for some constants $c>0$ and $R_0<\infty$ depending on the solution. In particular, gradient growth $o(R^{\alpha })$ forces flatness, resolving a conjecture of Mooney and the author. The result holds in every dimension, without any assumption that $\Phi$ is close to the Euclidean area integrand.

14 pages, 4 figures, comments welcome!