Ricci flow of the Torus

Question

Let us consider the torus of revolution, say $T$, and consider a local parametrisation

\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta. \end{aligned}

Given $T$ the induced metric of $\mathbb E^3$, we have in local coordinates $(\theta,\varphi)$ that we can write the metric as

\begin{aligned} g|_T=r^2d\theta^2+(R+r\cos\theta)^2d\varphi^2.\end{aligned}

I know that

\begin{aligned} Ric_{g|_T}=\frac {r\cos\theta}{R+r\cos\theta}d\theta^2+\frac 1r\cos\theta(R+r\cos\theta)d\varphi^2.\end{aligned} How can I, if it is even possible, find the explicit solution of the Ricci flow:

\begin{aligned} \partial_tg=-2Ric_g\\g(0)=g|_T~~~\end{aligned}

(I have seen how it is done for the sphere, cigar soliton, and the immortal solution stated in The Ricci Flow: An Introduction -- by Bennet Chow and Dan Knopf.)

If it is not possible, why? Thanks in advance.

Answer 1

Consider a point of negative curvature. Here metric is increasing so that $R,\ r$ goes to $\infty$. Hence a limit is flat metric.

Reference : Richard S. Hamilton. The Ricci flow on surfaces.