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I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. I'm curious about the possibilities for such a pinching in dimension 4, but I couldn't locate any information about it. I'm curious about 2 complementary questions in this regard.

  • Are there any known partial results or indications of what such a pinching may look like in dimension 4?
  • Are there known examples that constrain the form of or throw doubt upon such a possible pinching?

Any thoughts or references to the literature are welcome.

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2 Answers 2

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One striking example of the failure of Hamilton-Ivey pinching can be seen here in which it is shown that the FIK shrinkers (which do not have non-negative Ricci curvature, much less non-negative sectional curvature), can arise as blowup limits to the Ricci flow.

As far as I know, basically all that is known is that ancient solutions to the Ricci flow (e.g. blowup limits) have non-negative scalar curvature, by Corollary 2.5 here

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  • $\begingroup$ Thanks very much, this is the kind of thing I was looking for. I believe there should be a right parentheses after "sectional curvature" in your second line, and it should say "by Corollary 2.5" in the final line. The site wouldn't let me make the edits. $\endgroup$
    – Brian Klatt
    Commented Jun 27, 2014 at 15:40
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http://arxiv.org/pdf/0807.1582.pdf

In this paper, the author gave a local result of Hamilton-Ivey pinching on the gradient shrinking soliton ($n\ge 4$) with vanishing Weyl tensor (see prop 3.2). Also in the introduction part, the author announced a pinching result for the solution to Ricci flow in $n-$dimension provided the solution is LCF for all time, although I believe the proof has not been published yet.

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