Let $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$ be a cylinder with a flat product metric $g_{0}$. Denote by $\lambda_{0}>0$ the lowest eigenvalue of the torus $\mathbb{T}^{2}$. If $u$ is a harmonic function on $\mathbb{T}^{2}\times\mathbb{R}$ with growth control

(9.1)

$$|u(z)|=O(e^{\lambda z})$$

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