Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
@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
up vote 3 down vote accepted

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.

share|cite|improve this answer
Hmmm...I see, now. Thanks! – Ehsan M. Kermani Apr 6 '12 at 3:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.