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In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues ₁ (Bᵣ) of ``balls'' Bᵣ: =\₀ \ r² ₁ (Bᵣ) >0, \ we obtain a sharp estimate of the volume growth: |Bᵣ| cr^{ (). Moreover when >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 ²2 () >0. In particular, when M²>4, M is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.
Carron et al. (Wed,) studied this question.