Dear Dan: It’s true that the maximum principle I established in Lecture 3 is restricted to the compact case. It turns out that one can extend the maximum principle (in both scalar and tensor forms) to complete manifolds of bounded curvature by a barrier method (basically, one subtracts a small multiple of a smooth weight that grows exponentially at infinity) but it is somewhat technical (see e.g. Chapter 12 of “The Ricci Flow… part II” by Chow et al.). To oversimplify a little, maximum principles continue to hold in this case as long as the solution does not grow super-exponentially fast at infinity.

It might also be possible to establish the curvature bound by a Gronwall (or ODE comparison) type argument: starting with the heat equation one should be able to obtain an integral inequality of the form from the -boundedness of the heat propagator on tensors, and ODE comparison would then do the rest. Of course this is in some sense the maximum principle in disguise, since that is the easiest way to establish the above boundedness property, but if one has another way to get a handle on the heat propagator then one might possibly be able to avoid the maximum principle altogether.

]]>For example, in proving Proposition 1 you use a maximum principle argument for the reduced length, but I think this is non-problematic since the reduced length is big near infinity.

]]>a larger ball) can be viewed as a simpler version of some results in Section 6 of

Perelman’s second paper, where Ricci flow with surgery is considered. This material

is concerned with the long time behavior of Ricci flow with surgery. Hence it is

crucial in the proof of the geometrization conjecture. For the Poincare conjecture,

however, it is not necessary, which is probably why Morgan-Tian do not discuss it. ]]>