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

We prove that entire solutions of the minimal hypersurface equation $$\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{Du}{\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}}\right)=0$$ on a complete manifold with $\mathrm{R}\mathrm{i}\mathrm{c}\ge 0$, whose negative part grows like $\mathcal{O}(r\mathrm{/}\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$.