TITLE: Ricci flow with surgery in dimension 2
QUESTION [9 upvotes]: 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.

REPLY [13 votes]: 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 $h$.