# Does normalized Ricci flow on surfaces yield a bundle?

As is well known, the normalized Ricci flow is defined for all $t>0$ on compact surfaces, and every metric on a compact surfaces converges to a metric constant curvature if $X \neq S^2$ (at least I can't find a reference that asserts that same result for $X=S^2$; B. Chow's "Ricci flow on the 2-sphere" only shows that metrics of positive Gaußian curvature converge to constant curvature metrics). This is somewhat related to this.

One thus has a map $\mathcal{R}(X) \rightarrow T_X$ from the Riemannian moduli space to the Teichmüller space $T_X$ of constant curvature metrics associating to $g\in \mathcal{R}(X)$ its limit $g^\ast$ under the normalized Ricci flow. Note that the fibres of this map are convex.

Question: Is this map a fibre bundle?

• The fact that one has convergence to the round metric on the sphere follows from Chow's work, maybe not the paper you mention though. Precise references are to be found in Chow and Knopff book, "The Ricci flow, an introduction". Mar 29, 2012 at 22:17
• You're wrong, the paper of Chow that you quote proves convergence for all metrics on $S^2$, see his Corollary 1.3. Mar 30, 2012 at 0:45
• Note that even to show that your map is continuous requires some highly nontrivial analysis! Mar 30, 2012 at 2:08

The formal reference for the result is C. Earle and J. Eells, "A fibre bundle description of Teichmüller theory", J. Differential Geom. Volume 3, Number 1-2 (1969), 19-43. The upshot is that there are two fibrations: One is the fibration of the space of Riemannian metrics $R(S)$ over the space $C(S)$ of conformal structures (where the fiber is the space of positive functions), the second is the principal fibration of the space of conformal structures over the Teichmüller space $T(S)$ of the surface $S$ (where the fiber is $Diff_0(S)$). Combining these two one gets the fibration $R(S)\to T(S)$. To see that the first is a fibration one usually fixes a reference Riemannian metric $g_0$, then for $g\in R(S)$ the function $Vol(g)/Vol(g_0)$ determines a trivialization of $R(S)\to C(S)$. Doing this using uniformization is much harder, and requires some machinery which became fully available only by 1950s. Note that the existence of a constant curvature metric gives only a set-theoretic section. One also needs continuity which is harder and does not follow from some proofs of the Uniformization Theorem. (For instance, it does not follow from the Koebe's or Caratheodory's proofs, or the proof using Green's functions on the universal cover.)