An upper bound for the first nonzero Neumann eigenvalue

Let $\mathbb{M}$ denote a complete, simply connected Riemannian manifold with sectional curvature $K_{\mathbb{M}}\le k$ and Ricci curvature ${\text{Ric}}_{\mathbb{M}}\ge (n-1)K$, where $k,K\in \mathbb{R}$. Then for a bounded domain $\mathrm{\Omega }\subset \mathbb{M}$ with smooth boundary, we prove that the first nonzero Neumann eigenvalue $\mu_{1}(\Omega) \leq $\mathcal{C}$ \mu_{1}(B_{k}®)$\mathrm{.}Here$B_{k}®$isageodesicballofradius$R > 0$ in the simply connected space form ${\mathbb{M}}_k$ such that vol$(\mathrm{\Omega })$ = vol$(B_k(R))$, and $\mathcal{C}$ is a constant which depends on the volume, diameter of $\mathrm{\Omega }$ and the dimension of $\mathbb{M}$.