I'm not an expert on Ricci flow but I believe the rough general reason for this is as follows. In dimension 2 the normalized Ricci flow gives the following evolution equation for scalar curvature $$ \frac{\partial R}{\partial t}=\Delta_t R+R(R-r) $$ Where $r=\frac{\int_MR}{vol M}=2\pi \chi(M)$. The analysis of this equation is closely linked (via maximum principle) to that of the ODE $\frac{\partial R}{\partial t}=R(R-r) $. The nonzero stationary point of the ODE is $R=r=const$ (which is also the limit of $R$ under the Ricci flow as $t\to\infty$); it is stable when $r<0$ and unstable when $r>0$. Thus for $r<0$ the behaviors of the ODE and the PDE agree as both the diffusion laplacian term and the ODE term work in the "same direction" toward the stationary solution. this This makes the convergence estimates in this case quite easy. In contrast for $r>0$ the laplacian term and the ODE term work in "opposite directions" (with the laplacian term ultimately winning) which makes the analysis in this case more delicate.
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I'm not an expert on Ricci flow but I believe the rough general reason for this is as follows. In dimension 2 the normalized Ricci flow gives the following evolution equation for scalar curvature $$ \frac{\partial R}{\partial t}=\Delta_t R+R(R-r) $$ Where $r=\frac{\int_MR}{vol M}=2\pi \chi(M)$. The analysis of this equation is closely linked (via maximum principle) to that of the ODE $\frac{\partial R}{\partial t}=R(R-r) $. The stationary point of the ODE is $R=r=const$ (which is also the limit of $R$ under the Ricci flow as $t\to\infty$); it is stable when $r<0$ and unstable when $r>0$. Thus for $r<0$ the behaviors of the ODE and the PDE agree as both the diffusion laplacian term and the ODE term work in the "same direction" toward the stationary solution. this makes the convergence estimates in this case quite easy. In contrast for $r>0$ the laplacian term and the ODE term work in "opposite directions" (with the laplacian term ultimately winning) which makes the analysis in this case more delicate. |
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