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$\DeclareMathOperator\Diff{Diff}$Setting: Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the supremum norm ($\|f\|:=\max_{x \in M}\, |f(x)|$).

Let $\mathcal{X}(M)\subset L(C^{\infty}(M),C^{\infty}(M))$ be the set of smooth functions on $M$ which we equip with the metric induced by the operator norm $\|\cdot\|_{op}$. Let $\Diff_0(M)$ denote the set of orientation-preserving diffeomorphisms on $M$ which we equip with the uniform metric $$ d_{\infty}(f,g):= \sup_{x\in M}\, d_g(f(x),g(x)). $$

Question

Let $\operatorname{Exp}:(\mathcal{X}(M),\|\cdot\|_{op})\rightarrow (\Diff_0(M),d_{\infty})$ mapping any vector field $V\in \mathcal{X}$ to a diffeomorphism $\varphi_V:M\to M$ given by $\varphi(x):=x_1^x$ where $ \dot{x}_t^x = V(x_t^x) \mbox{ and } x_0^x = x. $

Is the map $\operatorname{Exp}$ locally Lipschitz? If so, are there estimates over ``sufficiently nice'' compact subsets of $\mathcal{X}(M)$?

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  • $\begingroup$ Which metrics are you using? $\endgroup$
    – Moishe Kohan
    Commented 2 days ago
  • $\begingroup$ Ah I'm equipped each space with their uniform metrics. $\endgroup$
    – ABIM
    Commented 2 days ago
  • $\begingroup$ Then spell it out: There are infinitely many inequivalent uniform metrics in this setting. $\endgroup$
    – Moishe Kohan
    Commented 2 days ago
  • 2
    $\begingroup$ @ABIM: let $p$ be a point where $v \neq 0$, choose local coordinates where $v|_p = \frac{\partial}{\partial x^1}$. Let your functions $f_\epsilon$ be such that, in a neighborhood of $p$, it equations $f_\epsilon(x_1, x_2, \ldots, x_n) = \epsilon \sin( \epsilon^{-2} x_1)$. They converge to $0$ uniformly, but $v(f)|_p$ blows up. // Since you are in the Riemannian category, why not just take the (sup of) the Riemannian length of the vector field? $\endgroup$
    – Willie Wong
    Commented yesterday
  • 4
    $\begingroup$ Fundamentally your question is really just the question of Lipschitz dependence on parameters for ordinary differential equations, dressed up in some geometry language which are not that significant. If you have access to Henri Cartan's Differential Calculus textbook, this is quite elegantly treated in Chapter 2, section 1.10 and 1.11. // Hartman's book has a discussion of $C^1$ dependence in Chapter 2, but if you are careful you can trace through the proof and track down the Lipschitz dependence (which requires a bit less hypotheses). $\endgroup$
    – Willie Wong
    Commented yesterday

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