Type problem, the first eigenvalue and Hardy inequalities

Abstract

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manifold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We obtain several sharp results. First, if for all $r>r_0$ \[ r^2 λ_1(B_r)\ge γ>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{μ(γ)}.$ Moreover when $γ>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $Δρ^2\geq2μ(γ)>0$. In particular, when $\inf_MΔρ^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.

Theorems & Definitions (33)

• Remark 1
• Theorem 1.1
• Remark 2
• Corollary 1.2
• Remark 3
• Corollary 1.3
• Proposition 1.4
• Proposition 1.5
• Corollary 1.6: cf. Carron97
• Lemma 2.1