Type problem, the first eigenvalue and Hardy inequalities

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues λ₁(Bᵣ) of ``balls'' Bᵣ:=\ρ<r\ on a complete Riemannian manifold M as r→ +∞, where ρ is a Lipschitz continuous exhaustion function with |∇ρ|≤1 a.e. on M. We obtain several sharp results. First, if for all r>r₀ \[ r^2 λ_1(B_r)≥ γ>0, \] we obtain a sharp estimate of the volume growth: |Bᵣ|≥ crμ(γ). Moreover when γ>j₀²≈ 5.784, where j₀ denotes the first positive zero of the Bessel function J₀, then M is hyperbolic and we have a Hardy type inequality. In the case where r₀=0, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that Δρ²≥2μ(γ)>0. In particular, when infMΔρ²>4, M is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.