X X CONTENTS OF PART I OF VOLUME TWO issue in applying the Riemannian compactness theorem is showing that the limiting metric/solution is Kahler fortunately this is easily handled. Chapter 4. In this chapter we first give an outline of the proof of the Cheeger-Gromov compactness theorem for pointed Riemannian manifolds. The proof of the compactness theorem for pointed manifolds is rather tech- nical and involves a few steps. The main step is to define, after passing to a subsequence, approximate isometries ^ from balls B (Ok, k) in Mk to balls B (Ofc+i, k + 1) in Mk+i- The manifold .Moo is defined as the direct limit of the directed system \E^. Convergence to Moo and the completeness of this limit follows from ^ being approximate isometries. The ideas in the proof of the main step are as follows. In each of the manifolds Mk in an appropriate subsequence, starting with the origins Ok = #£, one constructs a net (sequence) of points {Xk}a=o which will be the centers of balls B% of the appropriate radii (technically one considers balls of different radii for what follows). By passing to a subsequence and appealing to the Arzela-Ascoli theorem repeatedly, we may assume these Riemannian balls have a limit as k » oo for each a. Furthermore, we may also assume that the balls cover larger and larger balls centered at Ok and that the intersections of balls B% and B^ are independent of k in the limit. Choosing frames at the centers of these balls yields local coordinate charts H% (this depends on a decay estimate for the injectivity radius and our choice of the radii of the balls) and we can define overlap maps J^ \H%) ° H%. By passing to a subsequence, we may assume the J^ converge as k oo for each a and (3. The local coordinate charts also define maps between manifolds by F^ = Hf o(H^)'1. Now we can define approximate isometries Fki : B {Ok, k) Mi by taking a partition of unity and averaging the maps F£t Technically, this is accomplished using the so-called center of mass and nonlinear averages technique. This brings us to the remaining step, which is to show that these maps are indeed approximate isometries. Chapter 5. In Volume One we saw the integral monotonicity formula for Hamilton's entropy for solutions to the Ricci flow on surfaces with pos- itive curvature. There we also saw various curvature pinching estimates, the gradient of the scalar curvature estimate, and higher derivative of cur- vature estimates. Other monotonicity-type formulas, for the evolution of the lengths and areas of stable minimal geodesies and surfaces, yielded in- jectivity radius estimates in various special cases in low dimensions. In a generalized sense, all of these estimates may be thought of as monotonicity formulas. In this chapter we address Perelman's energy formula. One of the main ideas here is the introduction of an auxiliary function, which serves several purposes. It fixes the volume form, it satisfies a backward heat-type equa- tion, it is used to understand the action of the diffeomorphism group, and it relates to gradient Ricci solitons. We discuss the first variation formula for the energy functional and the modified Ricci flow as the gradient flow for
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