Let $(X^{2n},g)$ be a complete Riemannian manifold of dimension $2n$ which has non-negative Ricci curvature and is $\delta$-asymptotically Calabi for some $\delta>0$. Then there exists a $\epsilon_{X}>0$ depending on $(X^{2n},g)$ such that if $u$ is a harmonic function on $(X^{2n},g)$ satisfying

(5.3)

$$|u|=O(e^{\epsilon_{X}\cdot z^{\frac{n}{2}}}),\ z\to+\infty,$$

then $u$ is a constant.