4-dimensional Frobenius manifolds and Painlevé VI (Q476764)

In the early 90's B. Dubrovin observed that there is a bi-Hamiltonian hierarchy associated to any Frobenius manifold. In the paper under review the author extends the construction of Dubrovin to a tri-Hamiltonian hierarchy imposing however additional constraints on the Frobenius manifold. The author proves that 4-dimensional Frobenius manifolds having a tri-Hamiltonian structure are in one-to-one correspondance with the solutions to the Painlevé VI equation. In his textbook B. Dubrovin has shown that the solution to the Painlevé VI equation defines a 3-dimensional Frobenius manifold. Using these two results, where the same equation plays a completely different role, the author describes explicitly the procedure of constructing all tri-Hamiltonian 4-dimensional Frobenius manifolds from the 3-dimensional Frobenius manifolds.