A growth gap for anisotropic minimal graphs

Abstract

We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each $n\geq2$ 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 $\sup_{B_R^n}|Du|\geq cR^\alpha$ for all $R\geq R_0$, 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.