Geometry of shrinking Ricci solitons

Ovidiu Munteanu, Jiaping Wang

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

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 languageEnglish (US)
Pages (from-to)2273-2300
Number of pages28
JournalCompositio Mathematica
Volume151
Issue number12
DOIs
StatePublished - Dec 15 2015

Keywords

  • Ricci solitons
  • curvature estimates
  • diameter

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