Hyperbolic $p$-barycenters, circumcenters, and Moebius maps

Given a Moebius homeomorphism $f:\mathrm{\partial }X\to \mathrm{\partial }Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{{\mu }_x{\}}_{x\in X}$ on $\mathrm{\partial }X$, we describe a continuous family of extensions ${\hat{f}p : X \to Y }{1 \leq p \leq \infty}$of$f$,calledthehyperbolic$p$-barycentermapsof$f$. If all the measures ${\mu }_x$ have full support then for $p=\mathrm{\infty }$ the map ${\widehat{f}}_{\mathrm{\infty }}$ coincides with the circumcenter map $\widehat{f}$ defined previously in \cite{biswas5}. We use this to show that if $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$,thenthecircumcentermapsof$f$and$f^{-1}$are$\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. It follows that closed negatively curved manifolds with the same marked length spectrum are bi-Lipschitz homeomorphic.