# What will happen when $T^3$ evolves under Ricci flow?

At some day ago, I want to compute a specific example about Ricci flow. So, I try to compute the Ricci flow on $S^2\times S^2$, accord to the parameter of it. (I think the parameter is right, but seemingly not.)

In fact, I have computed out the Ricci tensor, connection, and balabla by software. At last, I get a system of nonlinear PDE, I really don't know how to get the solution. Besides, it has become very complex. So, I fail.

But I still want a non trivial example of Ricci flow, I think $T^3$ is a good start. But if I do it as before, I guess that, eventually, I will get a system of nonlinear PDE too.

So, I want to know, when studying Ricci flow on some manifold, what is the probable way? If I want to know the behaviour of $T^3$ under Ricci flow, what should I do?

• $T^3$ with the standard metric is flat, and hence is a stationary solution to Ricci flow. If you want a "nontrivial" example you shouldn't start with $T^3$. – Willie Wong Aug 22 '16 at 17:56
• Looking at your previous question, however, you were looking at $S^2\times S^2$ with a non-standard metric. Are you doing so for $T^3$ too? Ricci flow is an intrinsic geometric flow, and you need to specify the metric that you are using initially before you can compute anything. – Willie Wong Aug 22 '16 at 18:01