We show that the spectral gap of the Dirichlet form on the path space Px(M )T =C ([0,T ]→M ; γ(0)=x ) goes to 0 exponentially, when T → ∞. Here, M is a compact negatively curved manifold. This contrasts with the case of positive curvature. It is proved by using a gradient estimate of bounded harmonic functions on negatively curved manifolds.
