  We develop a general technique for solving the Riemann-Hilbert problem in
presence of a number of heavy charges and a small one thus providing the exact
Green functions of Liouville theory for various non trivial backgrounds. The
non invariant regularization suggested by Zamolodchikov and Zamolodchikov gives
the correct quantum dimensions; this is shown to one loop in the sphere
topology and for boundary Liouville theory and to all loop on the pseudosphere.
The method is also applied to give perturbative checks of the one point
functions derived in the bootstrap approach by Fateev Zamolodchikov and
Zamolodchikov in boundary Liouville theory and by Zamolodchikov and
Zamolodchikov on the pseudosphere, obtaining complete agreement.
