Flows on the $$\mathbf{PGL(V)}$$-Hitchin Component

Abstract

In this article we define new flows on the Hitchin components for $$\mathrm {PGL}(V)$$. Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when $$n=2$$. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the $$\mathrm{PGL} ({V})$$-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component.