On shrinking gradient Ricci solitons with nonnegative sectional curvature

Mingliang Cai

Vol. 277 (2015), No. 1, 61–76

DOI: 10.2140/pjm.2015.277.61

Abstract

Perelman proved that an open $\mathfrak{3}$-dimensional shrinking gradient Ricci soliton with bounded nonnegative sectional curvature is a quotient of $S^{\mathfrak{2}}\times ℝ$ or ${ℝ}^{\mathfrak{3}}$. We extend this result to higher dimensions with a decay condition on the Ricci tensor.

Keywords

shrinking gradient Ricci soliton, rigidity, nonnegative sectional curvature

Mathematical Subject Classification 2010

Primary: 53C25

Secondary: 53C20, 53C24

Milestones

Received: 21 March 2014

Revised: 23 December 2014

Accepted: 24 December 2014

Published: 6 August 2015