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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.

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    $\begingroup$ Probably the right version should involve instead a punctual curvature condition given by some measure, so that for instance, it would encompass the fact that in the plane, the sum of $|\pi-\alpha_i|$ for $\alpha_i$ the angles of a closed polygonal path, is at least $2\pi$. The curvature is naturally bounded by some "measure", which in this case is supported by the vertices, while it it non-atomic for a $C^1$-curve. I'm not sure how to define this properly, but say for a curve parameterized with speed one I'd copy the curvature definition and take the derivative in distribution sense. $\endgroup$
    – YCor
    Commented May 20, 2020 at 10:27
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    $\begingroup$ For the hyperbolic plane, a horosphere has constant geodesic curvature 1 and a circle of radius $r$ has constant geodesic curvature $\frac{1}{\tanh{r}}$, so I guess that for $X = \mathbb{H^2}$ we have $f(\epsilon) = \infty$ if $\epsilon < 1$ and $f(\frac{1}{\tanh{r}}) = \text{length}_{\mathbb{H}^2}(\partial B_r) = 2\pi \sinh(r)$. See math.stackexchange.com/questions/2430495/…. $\endgroup$
    – shurtados
    Commented May 20, 2020 at 17:34
  • $\begingroup$ I think this paper answers some versions of the question mathscinet.ams.org/mathscinet-getitem?mr=3415662 (not that $K \leq -1$ can be replaced by $K\leq 0$ if one makes the appropriate changes to the proof/statement). $\endgroup$
    – Otis Chodosh
    Commented Nov 23, 2022 at 0:56

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The Reshetnyak majorization theorem (see 9.56) 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.

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