In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality 
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact that the total curvature is $2|k|\pi$ with $k \not= 0$ the winding number. 


Is there a known generalisation of this to closed curves in a CAT(0) (simply connected with negative sectional curvature) Riemannian manifold? That is given such a manifold $X$, is there a function $f : [0, 1] \to [0, +\infty[$ (say) so that for any closed smooth curve in $X$ of length $\ell$ and curvature bounded above by $\epsilon$ we have $\ell \ge f(\epsilon)$? (of course we want $\lim\_{\epsilon\to 0} f(\epsilon) = 0$ and for pinched negative curvature I would expect it to go faster than $1/\epsilon$). 


We can also ask a purely metric version of this question: given a CAT(0)-space $X$, is there a function $f : [0, 1] \to [0, +\infty[$ (say) so that for any closed curve in $X$ of length $\ell$ which is a local $(1+\epsilon)$-quasi-geodesic we have $\ell \ge f(\epsilon)$? 


(By "local $(1+\epsilon)$-quasi-geodesic" I mean a curve $\gamma$ such that for any two points $x, y \in X$ such that $d(x, y) \le C$ and $x, y$ lie on $\gamma$, if $a$ is the length of the (shortest) arc between $x$ and $y$ on $\gamma$ then $a - d(x, y) \le \epsilon$. Here $C$ is a constant depending on $X$.)


**Edit following comments**: in Gromov-hyperbolic spaces it seems that the condition of being a "local-quasi-geodesic" implies (for sufficiently small $\epsilon$) that the curve is a global quasi-geodesic, in particular it cannot close. I think the proof of Theorem 1.13, p. 405 of Bridson--Haefliger can be immediately adapted to do this (and for this particular problem points (1) and (2) of the theorem are sufficient). 
So we can take $f = +\infty$ in a neighbourhood of 0 (depending on the hyperbolicity constant). The comment by shurtados shows that for the hyperbolic plane we can take the neighbourhood to be $[0, 1[$. 


As noted in Ycor's comment the "purely metric" version is not optimal for CAT(0) spaces (as opposed to hyperbolic) and for these a version that would include non-Riemannian $X$ and singular curves should probably involve some sort of "curvature measure" on the curve whose integral would be computable. This question seems interesting even for euclidean space. 



Length and curvature for closed curves in negatively curved spaces
The Reshetnyak majorization theorem (see [9.56](https://arxiv.org/pdf/1903.08539v5.pdf)) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be *majorized* by a convex plane figure $F$; that is, there is *short* (= 1-Lipschitz) map (= majorization) $m\colon F\to U$ such that $m|\_{\partial F}$ is the arc-length parametrization of $\alpha$.


Note that majorization does not decrease the curvature; that is, curvature of $\alpha$ cannot be smaller than curvature of $\partial F$ at the corresponding point.
Therefore, the inequality
$$ \ell \ge 2\cdot \pi\cdot \varepsilon^{-1}$$
holds in CAT(0) spaces as well.


