Type problem and the first eigenvalue

Abstract

In this paper, we study the relationship between the type problem and the asymptotic behavior of the first eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρ<r\}$ on a complete Riemannian manfold $M$ as $r\rightarrow +\infty$, where $ρ$ is a Lipschitz continuous exhaustion function with $|\nablaρ|\leq1$ a.e. on $M$. We show that $M$ is hyperbolic whenever \[ Λ_*:= \liminf_{r\rightarrow +\infty} \{ r^2 λ_1(B_r)\} >18.624\cdots. \] Moreover, an upper bound of $Λ_*$ in terms of volume growth $ν_*:=\liminf_{r\rightarrow +\infty} \frac{\log |B_r|}{\log r}$ is given as follows \[ {Λ_*} \lesssim \begin{cases} ν_*^2,\ \ \ &ν_*\gg1,\\ ν_*\log\frac{1}{ν_*},&1<ν_*\ll1. \end{cases} \] The exponent $2$ for $ν_*\gg1$ turns out to be the best possible.

Theorems & Definitions (35)

• Theorem 1.1
• Corollary 1.2
• Remark 1
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Proposition \ref{['prop:eigenvalue']}
• Theorem 2.1: cf. Grigoryan, Theorem 5.1
• proof : Proof of Theorem \ref{['th:ess']}