= An entire bounded vector field cannot have nonzero constant divergence
{title2=$\|X\|_\infty<\infty,\quad\operatorname{div}X=\kappa\ \Longrightarrow\ \kappa=0$}
For $X\in C^1(\mathbb R^n;\mathbb R^n)$, integration over $B_R$ gives $|\kappa||B_R|\leq\|X\|_\infty|\partial B_R|$, so $|\kappa|\leq n\|X\|_\infty/R$. 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 $Du$.
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