Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative curvature. I was wondering what happens to the ordinary Ricci flow, or if there is any condition under which one has long time existence? Thanks!

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    $\begingroup$ You might find it instructive to take a surface of constant negative curvature -1 and work out explicitly what happens for the normalized flow and the regular flow! $\endgroup$ – Otis Chodosh Sep 16 '15 at 6:40

On such a manifold, the flow will exist for all time starting from any initial metric. From applying the maximum principle to the evolution equation for the scalar curvature, the scalar curvature is bounded below for all time. By applying the maximum principle to a function like $\Delta f+|\nabla f|^2$ where $f$ is a solution of $\Delta f=R-\langle R\rangle$ with average value 0, Hamilton shows that the maximum scalar curvature will become negative in finite time. Since Hamilton also showed that the curvature of a Ricci flow has to blow up at a finite singularity time, the solution exists for all time.

After rescaling the flow it will converge to a constant curvature metric. It has to be rescaled since you can calculate that the area will increase at a constant rate due to the Gauss-Bonnet theorem.


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