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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.

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  • $\begingroup$ You might want to investigate the topic of Ricci flow with surgery as well as Ricci solitons. $\endgroup$
    – Deane Yang
    Commented Mar 9 at 16:15
  • $\begingroup$ Your notion of "small" is still too imprecise for your question to be answerable. Keep in mind that you can have a metric on, say, compact surface of genus 2 which has constant positive curvature outside of a coordinate disk. Same for each orientable compact 3-manifold. $\endgroup$
    – Moishe Kohan
    Commented Mar 9 at 16:24
  • $\begingroup$ Yes, a metric on the compact surface of genus 2 as you mentioned was in my mind that coordinate disk I mentioned as a small region with high negative curvature. $\endgroup$
    – Emmie
    Commented Mar 10 at 7:39
  • $\begingroup$ The answer to your first question is yes, examples being the metrics on higher genus surfaces discussed above. Ricci flow will take any metric on such a surface to a hyperbolic one (after parabolic blowing down). $\endgroup$
    – Ramiro Lafuente
    Commented Mar 11 at 3:12
  • $\begingroup$ Can you please elaborate a bit on the first question? Also about the second question 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 $𝑔$. $\endgroup$
    – Emmie
    Commented Mar 11 at 15:03

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I assume we are talking about a compact manifold $M$; then results from the geometry-to-topology paradigm can answer part of your questions.

The answer to your second question is negative, as there cannot exist both a non-positively curved metric and a positively curved metric on $M$ (let alone two such metrics in the same orbit of the Ricci flow). On the one hand, if there exist a non-positively curved metric $g$ on $M$, then $(M,g)$ is called a Cartan-Hadamard manifold, and the Cartan-Hadamard theorem states that $\tilde M$ is diffeomorphic to $\mathbb{R}^n$ (the exponential at any point providing a diffeomorphism). On the other hand, if there exist a positively curved metric $g_1$ on $M$, then the Myers theorem state that $\tilde M$ is compact (with a bound on its diameter in term of a lower bound on the Ricci curvature).

Regarding your first question, it depends on what one means by "mostly positive curvature". There could exist examples where there is a small region of negative curvature where curvature variations would have negative curvature spread out, but that seems very difficult to create without getting into singularities once one runs the Ricci flow.

My recollection is that Ricci flow behaves quite badly in negative curvature, and is more of a positive (Ricci) curvature thing.

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  • $\begingroup$ Thanks for your reply. No for that 2 nd question also, I now edited it, I wanted to ask if one starts with a metric $g$ with constant negative curvature can one get a metric $g_1$ always such that flowing $g_1$ via Ricci flow goves $g$ and $g_1$ is mostly positive curvature? So, by mostly positive curvature I mean it can have a small region with high negative curvature. $\endgroup$
    – Emmie
    Commented Mar 8 at 14:46
  • $\begingroup$ @Emmie I guess you mean to start from a negative curvature metric, not assumed to be constant (for constant curvature, the Ricci flow only dilateor contracts the metric, so curvature stays constant of the same sign). It would surprize me greatly if it would be possible to have negative curvature condense in a small region, but I am not knowledgeable enough to provide a proof (a possible keyword might be "maximum principle"). You should edit your question to include the precision on "mostly positive curvature", and the more precise you can explain how small the region should be the better. $\endgroup$
    – Benoît Kloeckner
    Commented Mar 9 at 13:57
  • $\begingroup$ I have edited it. Actually, I have intuitively some picture in mind where negative curvature condensed at some small part everywhere else the curvature is positive. Regarding how small it could be, it is not also intuitively clear to me that's why I asked this question here to get some idea. $\endgroup$
    – Emmie
    Commented Mar 9 at 15:47

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