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Keywords:
Hadamard manifold; Neumann eigenvalue; radial Ricci curvature
Hadamard manifold; Neumann eigenvalue; radial Ricci curvature
Summary:
Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k(t)$ with respect to some point $p\in M$, where $t=d(\cdot ,p)$ is the Riemannian distance on $M$ to $p$, $k(t)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty $, satisfy $$ \frac {1}{\mu _1}+\frac {1}{\mu _2}+\cdots +\frac {1}{\mu _n}\ge \frac {l^{n+2}}{(n+2)\int _{0}^{l}f^{n-1}(t){\rm d}t}, $$ where $f(t)$ is the solution to $$ \begin {cases} f''(t)+k(t)f(t)=0 \quad \text {on} \ (0,\infty ),\\ f(0)=0, \ f'(0)=1. \end {cases} $$
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