A Universal Bound for Lower Neumann Eigenvalues of the Laplacian

Abstract

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

$${1 \over {{\mu _1}}} + {1 \over {{\mu _2}}} + \ldots + {1 \over {{\mu _n}}}\geqslant {{{l^{n + 2}}} \over {(n + 2)\int_0^l {{f^{n - 1}}(t){\rm{d}}t} }},$$

where f(t) is the solution to

$$\left\{ {\matrix{ {f(t) + k(t)f(t) = 0\;\;\;\;{\rm{on}}\;(0,\infty ),} \hfill \cr {f(0) + 0,\;\;f\prime (0) = 1.} \hfill \cr } } \right.$$