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Is it possible to define the Ricci flow with surgery in dimension 2 and use it to classify the surfaces?

I know this is overkill, there are simpler ways to classify surfaces, but I would like to understand the Ricci flow with surgery in dimension 3 and perhaps that this is simpler in dimension 2.

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I'm definitely not an expert, but as far as I know you never need surgery in dimension 2. If you start with a compact (probably oriented, as well) 2-manifold, the Ricci flow always has long-time existence and convergence to a constant curvature metric. This does indeed give (I believe) the classification of compact oriented surfaces. –  Spiro Karigiannis Dec 2 '10 at 2:36
You can find this in the book of Ben Chow and Dan Knopf, by the way. –  Spiro Karigiannis Dec 2 '10 at 2:36
To add to what Spiro said, the long-time existence result is for the volume-renormalized Ricci flow. Otherwise the sphere is a shrinking soliton and there is no convergence of the flow. See also arxiv.org/abs/math/0505163 –  Willie Wong Dec 2 '10 at 3:28
While one doesn't need surgery to prove the uniformization theorem, there is some issue classifying the solitons in dimension 2. My understanding (also not an expert) is that for a long time the proofs that the only soliton metric on $S^2$ was the round sphere used the uniformization theorem. Recently, Chen, Lu and Tian proved this, though I don't know about other topologies. –  Rbega Dec 2 '10 at 3:32
@RBega: that's the arxiv paper I linked to above :-p. –  Willie Wong Dec 2 '10 at 3:43

3 Answers 3

up vote 16 down vote accepted

(1) The normalized Ricci flow (NRF) on compact surfaces always exists for all time and does not have singularities. Moreover, NRF fixes the conformal class of the metric.

(2) Let $r$ be the integral of the curvature, which is constant under any flow. Hamilton and Osgood-Phillips-Sarnak (independently, I think) showed that if $r\leq 0$, then the NRF converges to a metric of constant curvature. Hamilton also proved that if the curvature is positive everywhere, then NRF converges to a metric of constant curvature. This part of the argument unfortunately assumes the uniformization theorem.

(3) Chow showed that if $r>0$, then eventually the curvature will become positive everywhere and then Hamilton's argument applies.

(4) Much later, Chen, Lu, and Tian wrote a 2 page paper explaining how to remove the uniformization theorem from Hamilton's argument. http://arxiv.org/abs/math/0505163

(5) Putting it all together, Ricci flow gives an new proof of uniformization, i.e. every metric on a compact surface is conformal to a metric of constant curvature. Since the only constant curvature metrics are quotients of the standard sphere, Euclidean space, and hyperbolic space, one can then deduce the classification of surfaces.

As mentioned by Spiro, this whole story, except for (4), is told in the book by Chow and Knopf.

Note: A lot of this information is already in comments by others, but not in answer form. I'm writing this mainly because the accepted answer does not seem complete to me.

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Thank you, this is rather this kind of answer I wanted. –  Guillaume Brunerie Dec 9 '10 at 19:06

The Ricci flow in dimension two is (in essence) the gradient flow of the "Polyakov action" (renormalized $\log \det \Delta$). B. Osgood, R. Phillips, and P. Sarnak proved in the late eighties (using Polyakov's trace formula) that $\log \det \Delta$ is convex on conformal classes, the critical points are metrics of constant curvature, the function is proper (which also shows that isospectral sets of metrics are compact -- the most celebrated corollary of their result at the time), and hence the uniformization theorem follows. As pointed out in the previous comments, no surgery is necessary in dimension two.

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Where is this documented? Is it in one of Ben Chow's books? I confess I didn't know this. –  Deane Yang Dec 2 '10 at 4:40
I assume that the only part you don't know is that Ricci flow is the gradient flow of log det. For this, you can look at a (one page) paper by Kokotov and Korotkin Letters in Mathematical Physics (2005) 71:241–242 though the observation was known to Sarnak (and Hamilton, I would imagine) long before. I haven't read Chow's books, but I don't believe it is there (do correct me if I am wrong...) –  Igor Rivin Dec 2 '10 at 6:42
@Deane: you may also be interested in the recent work of Albin, Aldana, and Rochon extending the analysis to the non-compact case. That the Laplacian has continuous spectrum and the volume can be infinite gives rise to some interesting technical restrictions. See math.jussieu.fr/~albin (I'd link to arXiv, but I think arXiv is broken at the moment: I cannot get any abstract to open up.) At the very least, their paper also contains references to the facts Igor Rivin mentioned. –  Willie Wong Dec 2 '10 at 12:00

Regarding (3) in Dan Lee's answer, you don't need my work in the Ricci flow approach to the differential geometric version of the uniformization theorem. The reason is as follows. Take any metric $h$ on a closed orientable surface $M$ with $\chi (M) > 0$. Let $r$ denote the average scalar curvature of $h$, which is positive since $\chi$ is. Solve the equation $\Delta_h u = R_h - r$, where $R_h$ denotes the scalar curvature of $h$. This is possible since the integral of $R_h - r$ with respect to the measure induced by $h$ is zero. Define the pointwise conformally equivalent metric $g=e^uh$. Then we have $R_g=e^{-u}(-\Delta_h u + R_h) = e^{-u}r>0$. We can now use $g$ as an initial data for the normalized Ricci flow and apply Hamilton's theorem to obtain exponential convergence in any $C^k$-norm to a constant curvature metric in the same conformal class as $h$.

By now there are many approaches to the Ricci flow on closed surfaces, such as the Aleksandrov reflection method employed by Bartz--Struwe--Ye (inspired by Schoen's work on Yamabe (constant scalar curvature) metrics; in PDE see Serrin and Gidas--Ni--Nirenberg, etc.), Hamilton's isoperimetric monotonicity, application of Perelman's entropy formula, Andrews--Bryan's isoperimetric profile monotonicity, etc.

Regarding Igor Rivin's answer, it's been a long time since I've looked at this, but this is what I remember (please correct me if I am wrong). Osgood--Phillips--Sarnak looked at the Polykov action from the spectral point of view and also from the variational point of view. Aside: Ray--Singer's work on analytic torsion predates their work (I should also mention McKean, etal.). However they did not explicitly make the connection to Ricci flow, although they may have known this. They also did not reprove the differential geometric version of the uniformization in the positive Euler characteristic case since their proof of sequential convergence assumed conformality to the standard $S^2$. In my 2-sphere paper, I actually used the Polykov energy and the fact that it is bounded from below (I believe due to Onofri), which I learned from Osgood--Phillips--Sarnak (I am indebted to Richard Melrose for asking that I read this paper when I first arrived at MIT). I used the Polykov energy to control Hamilton' entropy in the variable signed curvature case. However, later in my entropy on 2-orbifolds paper, I found a way to avoid Polykov entropy to control the modified Hamilton's entropy. Yet another proof of the entropy bound, adapting the original Hamilton's contradiction argument, was in my paper with Lang-Fang Wu on 2-orbifolds with variable signed curvature.

Regarding some sort of convexity of the functional, a fancy way to interpret the energy functional is in terms of Bott--Chern secondary characteristic classes and this was originally used in Donaldson's work on Hermitian-Einstein metrics/Hermitian Yang-Mills connections on (semi-)stable vector bundles over algebraic surfaces (and later algebraic manifolds); Uhlenbeck--Yau had a different approach. In the case of closed Riemannian surfaces, the formula is: $$\ln \det \Delta_g - \ln \det \Delta_h = -\frac{1}{48\pi} E_h(g),$$ where the relative energy of two pointwise conformal metrics is defined by: $$E_h(g) = \int_M \ln (g/h)(R_g d\mu_g + R_h d\mu_h).$$ The Bott--Chern class is in effect the term $\ln (g/h)$ since $\partial \bar{\partial}$ of it is essentially the difference in the first Chern classes (Gauss--Bonnet integrands in this case) of $g$ and $h$.

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Welcome to MO ! –  BS. Oct 13 '13 at 19:36

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