A Riemann-Hilbert approach to Painleve IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painleve equation. One obtains a Riemann-Hilbert correspondence between moduli spaces of rank two connections on P-1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto-Painleve varieties and the Painleve property follows. For an explicit computation of the full group of Backlund transformations, rank three connections on P-1 are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.