Abstract: This is joint work with Anna Wienhard and Tengren Zhang. ) Let S be a closed, connected, oriented surface of genus at least 2. It is well-known that on Teichmuller space, the twist flows along a pants decomposition of S is a maximal family of Poisson commuting Hamiltonian flows. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the PSL(n,R) Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of the associated Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us compute the Goldman symplectic pairing explicitly. As a consequence, we find a maximal family of Poisson commuting Hamiltonian flows on the PSL(n,R) Hitchin component and a global Darboux coordinates for PSL(n,R) Hitchin component.
