An entire bounded vector field cannot have nonzero constant divergence ($\Vert X{\Vert }_{\infty }<\infty ,\mathrm{div}X=\kappa ⟹\kappa =0$)

For $X\in C^1({\mathbb{R}}^n;{\mathbb{R}}^n)$, integration over $B_R$ gives $|\kappa ||B_R|\le \Vert X{\Vert }_{\infty }|\partial B_R|$, so $|\kappa |\le n\Vert X{\Vert }_{\infty }/R$. Let $R\to \infty$. In particular, the bounded flux $Du/\sqrt{1+|Du{|}^2}$ of an entire constant-mean-curvature graph forces its divergence constant to be zero, independently of any bound on $Du$.