  For every non-elementary hyperbolic group, we introduce the Manhattan curve
associated to any pair of left-invariant hyperbolic metrics which are
quasi-isometric to a word metric. It is convex; we show that it is continuously
differentiable and moreover is a straight line if and only if the corresponding
two metrics are roughly similar, i.e., they are within bounded distance after
multiplying by a positive constant. Further, we prove that the Manhattan curve
associated to two strongly hyperbolic metrics is twice continuously
differentiable. The proof is based on the ergodic theory of topological flows
associated to general hyperbolic groups and analyzing the multifractal
structure of Patterson-Sullivan measures. We exhibit some explicit examples
including a hyperbolic triangle group and compute the exact value of the mean
distortion for pairs of word metrics.
