First Dirichlet eigenvalue $\lambda_1(B(z,R))$ in the hyperbolic space

Let us consider the hyperbolic space $\mathbb{H}^n$. Given $z \in \mathbb{H}^n$ and $R > 0$ let us consider the bottom eigenvalue $\lambda_1 = \lambda_1(B(z,R))$ of the Neumann problem $$ \Delta u + \lambda u = 0 \quad \text{and} \quad \frac{\partial u}{\partial \nu} _ {\partial B(z,R)} = 0. $$

I read that the first Neumann eigenvalue $\lambda_1$ decays exponentially. I do not find a reference for this.