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We prove that entire solutions of the minimal hypersurface equation \[  \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a completemanifold with $\mathrm{Ric} \ge 0$, whose negative part grows like$\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant.This extends the Bernstein Theorem for entire positive minimal graphsestablished in recent years. The proof depends on a new technique to getgradient bounds by means of integral estimates, which does not require anyfurther geometric assumption on $M$.

On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature