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