Geometry of shrinking Ricci solitons

Abstract

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 of M satisfies the estimate Rm ≤ cS for some constant c. Moreover, the curvature operator 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.