I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not.

So my question is if one starts with a metric that has mostly positive curvature and flows this via Ricci flow, can one end up getting a metric that has constant negative curvature? Also, the converse, i.e., if one starts with a metric with negative curvature say $g$ can one get some metric with mostly positive curvature say $g_1$ such that flowing $g_1$ via Ricci flow gives $g$? Here by curvature, I mean sectional curvature. Also the term 'almost positive' curvature means that there could be a small region with high negative curvature but rest at all places it's of positive curvature. The term 'small' means that we can allow a portion of the manifold that is relatively much smaller than the complement region where the curvature is highly negative.

Any insight on this will be very helpful for understanding purposes.

5more comments