Good afternoon everyone !


I have the following question of Riemannian geometry :


Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth Riemannian metric on $M$ with sectional curvature pinched between $-1- \epsilon$ and $-1$ $\}$ where $\epsilon$ is an arbitrary positive number. Assume that $\mathcal{T}$ is non-empty.


Let $\gamma $ be a simple closed curve in $M$. It is classical that for every negatively curved metric $g$ there is a unique closed curve in the free homotopy class of $\gamma$ that is length minimizing. Note $L\_g(\gamma)$ the length of such a curve.


1) Is it known whether $ \inf\_{g \in \mathcal{T} }{L\_g(\gamma)}$ is positive or zero ?


2) Is it known whether $ \sup\_{g \in \mathcal{T} }{L\_g(\gamma)}$ is finite or infinite ? 


3) Can one say more in specific cases, say when $M$ is hyperbolic ?


Obviously in dimension 2 Fenchel Nielsen coordinates show that $ \inf\_{g \in \mathcal{T} }{L\_g(\gamma)}$ is zero and $ \sup\_{g \in \mathcal{T} }{L\_g(\gamma)}$ is infinite. Nonetheless, the lack of topological symmetries for higher-dimensional hyperbolic manifolds make me hope that the opposite might be true.


Thanks for your attention !



Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
We have $0<\inf\_{g\in \mathcal{T}} L\_g(\gamma) \leq \sup\_{g\in \mathcal{T}} L\_g(\gamma) <\infty$. In fact, there should be a universal bound on the ratio
$\sup\_{g\in \mathcal{T}} L\_g(\gamma)/ \inf\_{g\in \mathcal{T}} L\_g(\gamma)$ for all $\gamma \in \pi\_1 M$.


This follows from a [theorem of Belegradek](http://www.ams.org/journals/proc/1999-127-04/S0002-9939-99-04654-7/home.html), who proves that the class of such metrics (actually, with just a fixed fundamental group $\pi$) is precompact in the Lipschitz topology. In particular, all such metrics are uniformly bi-Lipschitz, and thus one has the bound on the ratio between maximal and minimal lengths, as well as absolute bounds. 


Quoting from the paper: 
Recall that the class of all compact Riemannian manifolds of a given dimension
has the so-called Lipschitz topology, namely, two manifolds $M$ and $N$ are said to be
$\epsilon$-close if there exists a diffeomorphism $f : M → N$ such that both $f$ and $f^{−1}$ are
$e^\epsilon$-Lipschitz. A class of manifolds is called precompact if for any positive $\epsilon$, every
sequence of manifolds in the class has a subsequence whose members are mutually
$\epsilon$-close. 


Now, suppose there is no upper bound $C$ so that any two metric $g,h\in \mathcal{T}$ are $C$-close. Take sequences $g\_i,h\_i\in \mathcal{T}$, such
that $g\_i$ and $h\_i$ are not $N\_i$-close, for a sequence $N\_i\to \infty$. Passing to subsequences, we may assume that $\{g\_i\}$ are mutually $\delta$-close for any $\delta>0$, and similarly for $\{h\_i\}$. But $g\_i$ is $\delta$-close to $g\_1$, which is $C$-close to $h\_1$ for some $C$ (since these are metrics on the same manifold), and which is $\delta$-close to $h\_i$ for all $i$. Thus, $g\_i$ is $2\delta+C$-close to $h\_i$ for all $i$, a contradiction. 


So we see that any two metrics in $\mathcal{T}$ are $C$-close for some $C$. This implies that $\sup\_{g\in \mathcal{T}} L\_g(\gamma)/ \inf\_{g\in \mathcal{T}} L\_g(\gamma)\leq e^C$ for all $\gamma \in \pi\_1 M$. Moreover, comparing to any fixed metric in $\mathcal{T}$, we see that $0<\inf\_{g\in \mathcal{T}} L\_g(\gamma) \leq \sup\_{g\in \mathcal{T}} L\_g(\gamma) <\infty$.


