Abstract:

A Ricci flow is a time-parameter family of riemannian metrics $g(t)$ on a manifold that satisfy the second order PDE $dg/dt = -2 Ric(g)$ involving the Ricci curvature of the metric. The explicit PDE depends on the coordinates chosen for the manifold. A gradient soliton is a RF that evolves in time by diffeomorphisms (thus it must satisfy more tight constraints) and are special solutions to the RF. Most times, solitons are studied from an intrinsic point of view, since no explicit expression exists in coordinates. We will show, however, that in some cases an appropriate choice of coordinates can turn the PDE into a first order vector ODE, that can be analysed with a phase portrait. This gives a quite explicit and visual description of the soliton metrics, and provides exhaustive classifications under some conditions.