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A way to formalize smoothness of a flow is to think of the spacetime, see e.g. here. Let's say we are flowing a compact manifold $M$. Which of the following is true?

  1. $g_{ij}$ is a $C^1$ in time and $C^\infty$ in the space coordinate.

  2. $g_{ij}$ is a $C^\infty$ function on the spacetime?

  3. $g_{ij}$ is a $C^1$ function on the spacetime?

I think 2 is correct but then I am confused about Shi's estimates for higher derivatives. If two nearby metrics are $C^\infty$ close then all derivatives of the curvature tensor are also close, but Shi's estimates blow up when $t\to 0$. What am I missing? Update: I see the source of my confusion about Shi's estimates: the constants there depend only on the dimension and bounds on curvature tensor, but are otherwise independent of the initial metric. So this does not contradict 2.

I suppose the answer is based on a general PDE principle. What is a good reference?

Finally, will anything change if we a flowing a non-compact manifold (on which Ricci flow exists and is smooth)?

Igor Belegradek
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1 Answers1

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#2 is correct. You can see for instance sections 4-6 of Hamilton's 1982 paper, where his Nash-Moser theorem is applied directly on the Fréchet space of smooth sections of the pullback of the symmetric 2-tensor bundle to $M\times[0,1]$.

On any manifold whatsoever, "elliptic/parabolic regularity" results will imply that a family of metrics of even low assumed regularity which satisfy the Ricci flow equation must be smooth in the #2 sense. This is a purely local question and so is covered by purely PDE results. A standard reference with various levels of assumed regularity is Lieberman "Second order parabolic differential equation."

Quarto Bendir
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