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In this paper we first show that the necessary condition introduced in ourprevious paper is also a sufficient condition for a path to be a geodesic inthe group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms.This applies with no restriction on $M$. We then discuss conditions whichguarantee that such a path minimizes the Hofer length. Our argument relies on ageneral geometric construction (the gluing of monodromies) and on an extensionof Gromov's non-squeezing theorem both to more general manifolds and to moregeneral capacities. The manifolds we consider are quasi-cylinders, that isspaces homeomorphic to $M \times D^2$ which are symplectically ruled over$D^2$. When we work with the usual capacity (derived from embedded balls), wecan prove the existence of paths which minimize the length among all homotopicpaths, provided that $M$ is semi-monotone. (This restriction occurs because ofthe well-known difficulty with the theory of $J$-holomorphic curves inarbitrary $M$.) However, we can only prove the existence of length-minimizingpaths (i.e. paths which minimize length amongst {\it all} paths, not only thehomotopic ones) under even more restrictive conditions on $M$, for example when$M$ is exact and convex or of dimension $2$. The new difficulty is caused bythe possibility that there are non-trivial and very short loops in $\Ham^c(M)$.When such length-minimizing paths do exist, we can extend theBialy--Polterovich calculation of the Hofer norm on a neighbourhood of theidentity ($C^1$-flatness)

Hofer's L?-geometry: energy and stability of Hamiltonian flows, part II