Is there a Killing vector field on a complete Riemannian manifold $M$ with finite volume that satisfies the condition


$$\displaystyle\liminf\_{r\rightarrow +\infty} \displaystyle\frac{1}{r} \displaystyle\int\_{ B(2r)/B(r) } |X| d\nu\_g > 0, $$


where $B(r)$ denotes the geodesic ball of radius $r$ and centre $p$, where $p$ is any point in $M$?



Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume?
Consider a warped product of the hyperbolic plane $H$ with a circle $S^1$,
where the length of the circle is rescaled by a function $f\colon H\to(0,1]$
that is radially symmetric around $o\in H$. Write $f(x)=f(r)$ where $d(o,x)=r$ by abuse of notation. A sphere at distance $r$ in $H$
has volume $2\pi\sinh r$, so $f$ needs to satisfy
$$\int\_0^\infty f(r)\,\sinh r\,dr<\infty\;.$$
For example, take $f(r)=\frac1{1+r^2\sinh r}$ and let $(M,g)$ be the resulting Riemannian manifold with $g=g^{\mathrm{hyp}}\oplus (f\,d\varphi)^2$, with $\varphi$ the coordinate on $S^1$.


Then $(M,g)$ has a rotational symmetry. The corresponding Killing $X$ field has length $\sinh r$, where $r$ is now the pullback of $r$ to $M$. Hence for large $R$, approximately
$$\int\_{B(2R)\setminus B(R)}|X|\,d\nu\_g\sim\int\_R^{2R}f(r)\,(\sinh r)^2\,dr
\sim\int\_R^{2R}\frac{\sinh r}{r^2}\,dr\;.$$


