Riemann–Hilbert Problem: positive results

Formulation of the problem and its tautological solution on an abstract holomorphic vector bundle

Meromorhic trivialization and Plemelj theorem (solvability of the problem if one of the monodromies is diagonalizable).

Invariant subbundles, (ir)reducibility of a regular connexion.

Lemma on too different orders. Bounds on the splitting type of a bundle with irreducible Fuchsian connexion.

Bolibruch–Kostov theorem: solvability of the Riemann–Hilbert problem for irreducible representations.
Reading: Sections 18A18D from the book (printing disabled).
Note: in the formulation of Theorem 18.12 (p. 322), the word “Fuchsian” is missing (see the uploaded file).
Comment by Sergei Yakovenko — Thursday, March 20, 2008 @ 9:47 