# Example of shrinking Ricci soliton

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?

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.