Complete gradient shrinking Ricci solitons not having non-negative curvature

Question

Are there examples of complete gradient shrinking Ricci solitons not having non-negative Ricci curvature?

Answer 1

My comment seemed to get jumbled, so here is an expanded version:

In 3-d, an ancient complete Ricci flow (so, in particular a shrinking soliton) will have non-negative sectional curvature. This follows from a result of Chen, http://arxiv.org/pdf/0706.3081.pdf, Corollary 2.4. (If the manifold has controlled geometry at infinity, this follows from earlier results like Hamilton-Ivey pinching).

However, in higher dimensions, Hamilton-Ivey style pinching estimates do not work. (*). Indeed, as your question asks, there are shrinkers with non-positive Ricci curvature. For example, the solitons constructed by Feldman-Ilmanen-Knopf in http://projecteuclid.org/euclid.jdg/1090511686 do not have non-negative Ricci curvature.

An interesting question is whether or not these can arise as singularity models for a compact Ricci flow, and this was answered in the affirmative by Maximo: http://math.stanford.edu/~maximo/Maximo%20-%20On%20the%20blow-up%20of%20four%20dimensional%20Ricci%20flow%20singularities.pdf. An interesting consequence of his work is that positive Ricci curvature is not preserved along the flow in higher dimensions (EDIT: as remarked the answer by @BewSMA, the lack of preservation of Ricci curvature under the flow was already observed).

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EDIT: I'll just point out that the above paper of Chen proves that a shrinker must have non-negative scalar curvature (see Corollary 2.5).

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(*) In spite of this aspect of the theory failing, there are curvature conditions which are preserved under the flow. For example, see http://www.ams.org/journals/jams/2009-22-01/S0894-0347-08-00613-9/S0894-0347-08-00613-9.pdf

Answer 2

For the positive Ricci curvature not preserved by Ricci flow, the first compact example(as I know) is given by Boehm and Wilking in the following paper(the example is 12 dimensional with positive sectional curvature initially):

Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geom. Funct. Anal. 17 (2007), no. 3, 665–681.

Complete examples that the non-negative sectional curvature is not preserved by Ricci flow are given by Lei Ni in the following paper:

Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11(2004), 883-904.

Ni's construction is quite interesting that uses Cheeger-Gromoll splitting. One may check if the non-negative Ricci curvature on his example is preserved or not.