# Hamilton-Ivey pinching in dimension 4

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.

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

• 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. Jun 27, 2014 at 15:40

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.