On the spectrum of cartan-hadamard manifolds

Let I be a simply-connected complete d-dimensional Riemannian manifold of nonpositive sectional curvature K.If K≦– k²<0, then the infimum of the L²spectrum of the negative Laplacian is greater than or equal to (d–1)²K²/4 with equality in case K⟶–k²sufficiently fast at infinity.This general result is obtained by analyzing a system of ordinary differential equations. If either d=2 or the manifold possesses appropriate symmetry, the result is obtained under weaker conditions by analyzing a Riccati equation.Finally the case k=0 is treated separately.