A NEW LAPLACIAN COMPARISON THEOREM AND THE ESTIMATE OF EIGENVALUES

This paper establishes a new Laplacian comparison theorem which is specially useful to the manifolds of nonpositive curvature. It leads naturally to the corresponding heat kernel comparison and eigenvalue comparison theorems. Furthermore, a lower estimate of $L^2$-spectrum of an $n$-dimensional non-compact complete Cartan-Hadamard manifold is given by $(n-1)k/4$, provided its Ricci curvature $\le-(n-1)k$ $(k=\roman {const.}\ge 0)$.