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Geometry of shrinking Ricci solitons

Part of: Global differential geometry

Published online by Cambridge University Press:  29 July 2015

Ovidiu Munteanu  and
Jiaping Wang
Ovidiu Munteanu
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06268, USA email ovidiu.munteanu@uconn.edu
Jiaping Wang
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email jiaping@math.umn.edu
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Abstract

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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.

Keywords

Ricci solitonscurvature estimatesdiameter

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2273 - 2300
Copyright
© The Authors 2015 

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