# Non unique solution for Ricci flow equation

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

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.