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 ${\lambda }_1(B_r)$ of “balls” $B_r:=\{\rho <r\}$ on a complete Riemannian manifold $M$ as $r\to +\infty$, where $\rho$ is a Lipschitz continuous exhaustion function with $|\nabla \rho |\le 1$ a.e. on $M$. We obtain several sharp results. First, if $r^2{\lambda }_1(B_r)\ge \gamma >0$ for all $r>r_0$, we obtain a sharp estimate of the volume growth: $|B_r|\ge c{r}^{\mu (\gamma )}.$ Moreover, when $\gamma >j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is non-parabolic 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 $\Delta {\rho }^2\ge 2\mu (\gamma )>0$. In particular, when ${\inf }_M\Delta {\rho }^2>4$, $M$ is non-parabolic and we get a sharp Hardy-type inequality. Related results for finite volume case are also studied.