Let $(M,g,X)$ be a shrinking Ricci soliton. Is it possible that the Ricci curvature $Ric$ satisfies the following inequality
$$Ric\_x(v)\leq \frac{C}{r}\quad \forall v\in T\_xM\text{ and } \forall x\in B(2r),$$
where $B(2r)$ is the geodesic ball with radius $r$ and center $o$ for a fixed point $o\in M$ and $C>0$ is a constant?



Example of shrinking Ricci soliton
Yes: Take the shrinking Gaussian $(\mathbb{R}^n, dx^2)$ with $X=\rho\nabla\rho$, where $\rho$ denotes the distance to the origin. This space is Ricci flat, so your inequality holds.


