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While reading Peter Topping's “Lectures on the Ricci Flow”, I came across the term “linearization” and was unsure of its precise meaning in this context. The book considers a nonlinear differential operator acting between spaces of smooth sections of a vector bundle over a compact manifold. I understand that linearization refers to a form of linear approximation. However, the space of smooth sections is a Fréchet space, not a Banach space, while the Fréchet derivative is typically defined for operators between Banach spaces. This leads me to the following questions:

  • In this context, does “linearization” refer to the Fréchet derivative of the operator once it has been extended to act between suitable Sobolev spaces? Or is it something defined more directly on the Fréchet space, perhaps related to the Gâteaux derivative? Are these approaches equivalent?
  • Is it possible to define a Fréchet derivative directly on Fréchet spaces? For instance, could one simply replace the norm in the definition with a seminorm? Why does this approach seem to be avoided? I suspect that for operators with good regularity, such a definition might coincide with the practical linearization used in the field, but I'm not sure why this isn't standard.

If anyone have any ideas, or have good references please let me know.

Thank you.

G.W.
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  • If you find any good references, post here as well. I wanted to understand Ricci flow. Topping's lectures are the most accessible IMO. I was trying to read it back when I had the insane desire to understand Perelman's work XD. – arjo Jun 21 '25 at 09:06
  • Thanks for your comment now I have clear idea what to learn to get rigorous understanding. So after learning about this topic I’ll shere some references. – G.W. Jun 22 '25 at 07:39
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    Also, If you want to learn about Ricci flow, I also recommend Chow, Knoph’s book. It contains some important topics or examples Topping omitted(e.g., proof of uniformization theorem of Riemannian surface using Ricci flow.) – G.W. Jun 22 '25 at 07:46

1 Answers1

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Linearization in Ricci Flow Context "Linearization" refers to the Gâteaux derivative computed on smooth sections, then extended to Sobolev spaces where it becomes a Fréchet derivative.

For operator $F: \Gamma(E) \to \Gamma(F)$: $$DF|_u(h) = \lim_{t \to 0} \frac{F(u + th) - F(u)}{t}$$

Why Sobolev spaces? Yes, Fréchet derivatives exist directly on Fréchet spaces using seminorm families instead of norms. But this approach is avoided because:

  1. No analytical power: Elliptic theory, compactness, Fredholm theory all live in Banach spaces
  2. Weak inverse function theorem: Fréchet space IFT has restrictive hypotheses
  3. No regularity theory: Can't bootstrap smoothness without Sobolev embeddings

The Gâteaux derivative on $C^{\infty}$ sections equals the Fréchet derivative on Sobolev spaces when both exist, but Sobolev spaces provide the tools needed for existence/uniqueness theorems.

You compute the linearization where it's natural (smooth sections), then study it where you have analytical tools (Sobolev spaces).

arjo
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