Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 473-482, 2020


A universal bound for lower Neumann eigenvalues of the Laplacian

Wei Lu, Jing Mao, Chuanxi Wu

Received August 7, 2018.   Published online December 13, 2019.

Abstract:  Let $M$ be an $n$-dimensional ($n\ge2$) 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 $\frac1{\mu_1}+\frac1{\mu_2}+\cdots+\frac1{\mu_n}\ge\frac{l^{n+2}}{(n+2)\int_0^lf^{n-1}(t){\rm d}t},$ where $f(t)$ is the solution to $f (t)+k(t)f(t)=0$ on $(0,\infty)$, $f(0)=0, f'(0)=1.$
Keywords:  Hadamard manifold; Neumann eigenvalue; radial Ricci curvature
Classification MSC:  35P15, 53C20
DOI:  10.21136/CMJ.2019.0363-18


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Affiliations:   Wei Lu, Jing Mao (corresponding author), Chuanxi Wu, Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, 368#, You Yi Road, Wuchang District, Wuhan 430062, Hubei Province, P. R.China, e-mail: jiner120@163.com


 
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