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An upper bound for the first nonzero Neumann eigenvalue
Journal of Geometry and Physics ( IF 1.6 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.geomphys.2020.103838
Sheela Verma

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 $\mathbb{M}$.

中文翻译:

第一个非零 Neumann 特征值的上限

令 $\mathbb{M}$ 表示一个完整的、简单连通的黎曼流形,其截面曲率为 $K_{\mathbb{M}} \leq k$ 和 Ricci 曲率 $\text{Ric}_{\mathbb{M}} \ geq (n-1)K$,其中 $k,K \in \mathbb{R}$。然后对于边界光滑的有界域 $\Omega \subset\mathbb{M}$,我们证明第一个非零 Neumann 特征值 $\mu_{1}(\Omega) \leq \mathcal{C} \mu_{1} (B_{k}(R))$。这里 $B_{k}(R)$ 是一个半径为 $R > 0$ 的测地球在简单连通空间形式 $\mathbb{M}_{k}$ 中,使得 vol$(\Omega)$ = vol$ (B_{k}(R))$,而$\mathcal{C}$是一个常数,取决于$\Omega$的体积、直径和$\mathbb{M}$的尺寸。
更新日期:2020-11-01
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