Let M be an n -dimensional ( n ⩾ 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 ∈ M , where t = d (·, p ) is the Riemannian distance on M to p, k ( t ) is a nonpositive continuous function on (0, ∞), then the first n nonzero Neumann eigenvalues of the Laplacian on the geodesic ball B ( p , l ), with center p and radius 0 < l < ∞, satisfy $${1 \over {{\mu _1}}} + {1 \over {{\mu _2}}} + \ldots + {1 \over {{\mu _n}}}\geqslant {{{l^{n + 2}}} \over {(n + 2)\int_0^l {{f^{n - 1}}(t){\rm{d}}t} }},$$ 1 μ 1 + 1 μ 2 + … + 1 μ n ⩾ l n + 2 ( n + 2 ) ∫ 0 l f n − 1 ( t ) d t , where f ( t ) is the solution to $$\left\{ {\matrix{ {f(t) + k(t)f(t) = 0\;\;\;\;{\rm{on}}\;(0,\infty ),} \hfill \cr {f(0) + 0,\;\;f\prime (0) = 1.} \hfill \cr } } \right.$$ { f ( t ) + k ( t ) f ( t ) = 0 o n ( 0 , ∞ ) , f ( 0 ) + 0 , f ′ ( 0 ) = 1.

中文翻译：

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拉普拉斯算子的下诺依曼特征值的普遍界

设 M 是一个 n 维 (n ⩾ 2) 简单连通的 Hadamard 流形。如果 M 的径向 Ricci 曲率从下方以 (n − 1) k ( t ) 为界，相对于某个点 p ∈ M ，其中 t = d (·, p ) 是 M 到 p, k ( t ) 是 (0, ∞) 上的非正连续函数，那么测地球 B ( p , l ) 上拉普拉斯算子的前 n 个非零 Neumann 特征值，以 p 为中心，半径为 0 < l < ∞，满足 $${ 1 \over {{\mu _1}}} + {1 \over {{\mu _2}}} + \ldots + {1 \over {{\mu _n}}}\geqslant {{{l^{n + 2}}} \over {(n + 2)\int_0^l {{f^{n - 1}}(t){\rm{d}}t} }},$$ 1 μ 1 + 1 μ 2 + ... + 1 μ n ⩾ ln + 2 ( n + 2 ) ∫ 0 lfn − 1 ( t ) dt ，其中 f ( t ) 是 $$\left\{ {\matrix{ {f(t) + k(t)f(t) = 0\;\;\;\;{\rm{on}}\;(0,\infty ),} \hfill \cr {f(0) + 0,\;\ ;f\prime (0) = 1.} \hfill \cr } } \right.