The Weak Maximum Principle

The maximum principle is the main tool we will use to understand the behaviourof solutions to the Ricci flow. While other problems arising in geo- metric analysis and calculus of variations make strong use of techniques from functional analysis, here – due to the fact that the metric is changing – most of these techniques are not available; although methods in this direction are developed in the work of Perelman [Per02]. The maximum principle, though very simple, is also a very powerful tool which can be used to show that pointwise inequalities on the initial data of parabolic pde are preserved by the evolution. As we have already seen, when the metric evolves by Ricci flow the various curvature tensors R, Ric, and Scal do indeed satisfy systems of parabolic pde. Our main applications of the maximum principle will be to prove that certain inequalities on these tensors are preserved by the Ricci flow, so that the geometry of the evolving metrics is controlled.