Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton M with bounded scalar curvature S , it is shown that the curvature operator Rm of M satisfies the estimate |Rm| cS for some constant c . Moreover, the curvature operator Rm is asymptotically nonnegative at infinity and admits a lower bound Rm -c(ln (r+1))-1/4 , where r is the distance function to a fixed point in M . As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.