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 !


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