An upper bound for the first nonzero Neumann eigenvalue

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

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