The Ricci flow on the 2-sphere

The Ricci Flow on Noncommutative Two-Tori

“…The properties of the Ricci flow in 2-dimension were studied in [33] and [13], [34], [38]. As shown in [33], in 2-dimension the Ricci flow is related to the gradient flow of the Yamabe problem.…”

“…For surfaces of genus g ≥ 1, in was shown in [33] that any metric flows to a constant curvature metric and in the genus zero case any metric with positive Gauss curvature also flows to a constant curvature metric. The latter assumption in the genus zero case was removed in [13] by showing that, for any metric on S 2 , the Gaussian curvature becomes positive in finite time under the Ricci flow. In particular, in the case of a commutative 2-dimensional torus, an arbitrary initial metric flows to the flat metric under the Ricci flow, [33].…”

On the weak Kähler-Ricci flow

“…Extensive research has been done in the case of the smooth flow (cf. [22], [6], [14], [21], [25], [12], [13], [30], or [15] for complete updated references). In order to prove the uniqueness of extremal Kähler metrics in full generality, in [9], the first two named authors were led to the study of Kähler-Ricci flows in the weak sense.…”

Ricci flow on surfaces with cusps

“…There are, of course, many different proofs of this classical theorem, including quite a few which rely primarily on PDE techniques. For proofs in the compact case using Ricci flow see [6], [4] and [2]; [5] has a proof using a fourth order flow. The paper [9] contains a PDE proof of the general uniformization theorem on arbitrary open Riemann surfaces.…”