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Geometry of shrinking Ricci solitons
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 12 / December 2015
- Published online by Cambridge University Press:
- 29 July 2015, pp. 2273-2300
- Print publication:
- December 2015
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- Article
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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 
$\text{Rm}$ of 
$M$ satisfies the estimate 
$|\text{Rm}|\leqslant cS$ for some constant 
$c$. Moreover, the curvature operator 
$\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound 
$\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, 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.
