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Why completeness is important for the uniqueness of solution to Ricci flow? For example, if $M$ is the open unit disk in $\mathbb{R}^2$ and $g(0)$ is the Euclidean metric, and hence not complete. Why the solution $g(t)$ to Ricci flow is not unique?

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@Ryan: sorry, I don't understand your question! –  Ehsan M. Kermani Apr 6 '12 at 1:20
    
I think I misunderstood your question. Okay now I see. I'll respond. –  Ryan Budney Apr 6 '12 at 1:55

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Say you have an incomplete Riemann manifold $M$, and for a moment imagine it embeds as an open subset of a complete Riemann manifold $N$. Then the Ricci flow on $N$ restricts to the Ricci flow on $M$, as Ricci flow is local. But you can change the metric on $N$ anyway you like, and the Ricci flow on $N$ may be different.

Think for example about the case of a flat ball -- you could embed it in a sphere with a flat open subset, the rest being more or less round. Or you could embed your flat ball in Euclidean space.

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Hmmm...I see, now. Thanks! –  Ehsan M. Kermani Apr 6 '12 at 3:28

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