Let $(Q,g_{Q})\equiv(D\times\mathbb{R},g_{Q})$ be a cylinder with a product Riemannian metric $g_{Q}=g_{D}\oplus dz^{2}$, where $(D,g_{D})$ is a closed Riemannian manifold. Denote by $\lambda_{D}>0$ the lowest eigenvalue of the Laplace-Beltrami operator of $(D,g_{D})$ acting on functions. If $u$ is a harmonic function on $Q$ satisfying the growth control

(6.59)

$$|u|=O(e^{\lambda_{c}\cdot z})$$

for some $\lambda_{c}\in(0,\sqrt{\lambda_{D}})$, then $u\equiv 0$.