Non unique solution for Ricci flow equation

Question

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?

Answer 1

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.