On page 12 of "Lectures On Ricci Flow" by Peter Topping is written:
In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working directly from the equation $\frac{\partial g}{\partial t}=-2Ric(g) $, we then see that regions in which $K < 0$ tend to expand, and regions where $K > 0$ tend to shrink.
Can anyone solve the Ricci flow PDE in this case and show regions in which $K < 0$ tend to expand, and regions where $K > 0$ tend to shrink?
thanks
Substituting $Ric=Kg$ into the Ricci flow equation we get $\dot g = -2Kg$, where $\dot g$ is the time derivative of $g$. Since $K$ is a scalar, this equation simply means that every component of $g$ satisfies the same equation (considering $g$ as a matrix): $\dot g_{ik}=Kg_{ik}$, where $i,k=1,2$. Hence, without loss of generality assuming that $g$ is a diagonal matrix, if $K<0$ then $\dot g_{ii}>0$, which means that the length of a vector, say $v$, will grow for a short time at least (more precisely, the time derivative of the length will be positive).