The $p-$parabolicity under a decay assumption on the Ricci curvature

Abstract

We prove that, given $\alpha >0$, if $M$ is a complete Riemannian manifold which Ricci curvature satisfies.\[\operatorname*{Ric}\nolimits_{x}(v)\geqα\operatorname{sech}^{2} (r(x)))\] or \[ \operatorname*{Ric}\nolimits_{x}(v)\geq-\frac{{h_α} (r(x))}{r(x)^{2}}, \] where \[ {h_α}(r) = \frac{α(α+1)r(x)^{α}}{r(x)^{α}-1}, \] for all $x\in M\backslash B_R(o)$ and for all $v\in T_{x}M,$ $\Vert v\Vert =1,$ where \ $o$ is a fixed point of $M$, $r(x)=d(o,x)$, $d$ the Riemannian distance in $M$ and $B_R(o)$ the geodesic ball of $M$ centered at $o$ with radius $R>0$, then $M$ is $p-$parabolic for any $p>1$, if satisfies the first inequality, and $M$ is $p-$parabolic, for any $p\ge (\alpha +1)(n-1)+1$, if satisfies the second inequality.