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Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function, \begin{align} L_{g}:\mathcal{M}_{-}\rightarrow \mathbb{R}, \end{align} where $\mathcal{M}_{-}$ is the space of Riemannian metrics on $M$ which have negative sectional curvature. The question is the following, suppose $g_{t}$ is a $C^k$-one parameter family of negatively curved Riemannian metrics defined on some open interval $(a,b).$ What is the regularity of $L_g:(a,b)\rightarrow \mathbb{R}$ defined by $t\mapsto L_{g}(g_t)?$ In an ideal world, if the variation in metrics is real analytic, this function would also vary real analytically. There are some known instances of this phenomenon. For instance, the paper of Bridgeman-Taylor shows the veracity of this result for the slightly different case of convex-cocompact hyperbolic metrics (which are infinite volume, but it's a very similar setup). https://www2.bc.edu/~bridgem/papers/wpextension.pdf

Unfortunately, their proof relies crucially on some properties of the deformation theory of convex-cocompact hyperbolic metrics, and so cannot be carried over to this context.

Some explanation now. For a pair of negatively curved metrics $g,g',$ there is a conjugacy (for the $\pi_1(M)$-action) of the space of geodesics in the universal cover $\widetilde{M},$ \begin{align} f:\mathcal{G}(g)\rightarrow \mathcal{G}(g'). \end{align} The Patterson-Sullivan geodesic current $\sigma_g$ (for g) is a measure on the space of geodesics $\mathcal{G}(g)$ which is invariant under the action of $\pi_1(M).$ Using $f,$ one obtains the push-forward measure $f_{*}(\sigma_g).$ Combining this measure with arc-length measure for $g'$ along geodesics gives a measure on the total space of an $\mathbb{R}$-bundle over the space of geodesics, \begin{align} T^{1}(\widetilde{M})\rightarrow \mathcal{G}(g'). \end{align} The map here sends a unit vector to the unique geodesic to which it is tangent. This picture is invariant under the $\pi_1(M)$-action, and taking the quotient by the action yields a measure on the unit tangent bundle $T^{1}(M).$ The function $L_g(g')$ records the total mass of this measure.

I didn't define the Patterson-Sullivan geodesic current here, but I'm banking on the fact that anyone who answers will already know the construction of this measure.

If anyone would like more detail, I'll be happy to write it up. Let me know.

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If $M$ is compact or convex-cocompact, this map should be at least Hölder continuous. Because the boundary at infinity is a well defined topological space with a Hölder structure, even if you change the metric. It is far from what you want, and does not use the analyticity of the variation. A good reference is Ledrappier, Structure au bord des variétés à courbure négative, in french.

If $M$ is a surface, I guess that the same kind of results as Bridgeman-Taylor's shoud hold.

On a related question, whose arguments can maybe help, the critical exponent varies analytically when the manifold is compact or convex-cocompact, and the variation of metrics is analytic, by results of Katok-Knieper-Pollicott-Weiss -Differentiability of entropy for Anosov and geodesic flows, and Tapie " A variation formula for the topological entropy of convex-cocompact manifolds". But all these results use coding argument. Without any compactness, you can hope $C^1$-regularity but it is difficult to expect more.

Barbara

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  • $\begingroup$ Thanks for your comment! I'm aware of these other results, but I haven't figured out how to make any headway with them yet. Indeed, at least continuity is fairly straightforward (it's the same argument as in B-T), and I'd certainly believe Holder continuous. $\endgroup$ – Andy Sanders May 6 '14 at 21:38

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