An upper bound for the first nonzero Neumann eigenvalue

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