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Positively curved shrinking Ricci solitons are compact. (English) Zbl 1405.53072

A complete Riemannian manifold \((M,g)\) is a shrinking gradient Ricci soliton if the equation \[ \text{Ric}+\text{Hess}(f)=\frac12 g \] holds for some function \(f\). In the work [arXiv e-print service , Paper No. 0303109, 22 p. (2003; Zbl 1130.53002)], G. Perelman proved that if \((M,g)\) is a three-dimensional non-collapsing gradient shrinking Ricci soliton with positive and bounded sectional curvature, then \((M,g)\) must be compact.
In [Chin. Ann. Math., Ser. B 27, No. 2, 121–142 (2006; Zbl 1102.53025)], H.-D. Cao posed the question whether this theorem is true in full generality in any dimension.
In the present paper the authors prove the following:
{Theorem}. Let \((M,g)\) be an \(n\)-dimensional gradient shrinking Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. Then \((M,g)\) must be compact.
By combining the above theorem with results of O. Munteanu and N. Sesum [J. Geom. Anal. 23, No. 2, 539–561 (2013; Zbl 1275.53061)] and P. Petersen and W. Wylie [Geom. Topol. 14, No. 4, 2277–2300 (2010; Zbl 1202.53049)], they obtain the following:
{Corollary}. Let \((M,g)\) be an \(n\)-dimensional gradient shrinking Ricci soliton with nonnegative sectional curvature. Then \((M,g)\) must be compact or a quotient of \(\mathbb{R}^n\) or of the product \(\mathbb{R}^k\times N^{n-k}\) with \(1\leq k\leq n-2\), where \(N\) is a compact simply connected shrinking Ricci soliton of dimension \(n-k\) with positive Ricci curvature.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)