## 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.

Original language | English (US) |
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Pages (from-to) | 2273-2300 |

Number of pages | 28 |

Journal | Compositio Mathematica |

Volume | 151 |

Issue number | 12 |

DOIs | |

State | Published - Dec 15 2015 |

### Bibliographical note

Publisher Copyright:© 2015 Foundation Compositio Mathematica.

## Keywords

- Ricci solitons
- curvature estimates
- diameter