The Ricci flow on the 2-sphere

The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. Thus one can con-sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met-ric. The primary interest in this question is not so much to give a newproof of the uniformization theorem, but rather to understand nonlinearparabolic equations better, especially those arising in differential geome-try. A sufficiently deep understanding of parabolic equations should yieldimportant new results in Riemannian geometry.The question in the preceding paragraph has been studied by RichardHamilton 3 and Brad Osgood, Ralph Phillips and Peter Sarnak 6. In 3,Hamilton studied the following equation, which we refer to as