CHAPTER 7
Derivative estimates
We saw in Chapter 3 that the Ricci flow is equivalent via DeTurck's trick
to a quasilinear parabolic PDE, and may in fact be regarded heuristically as
a nonlinear heat equation for a Riemannian metric. Furthermore, we saw
in Chapter 6 that the intrinsically defined curvatures of a Riemannian met-
ric evolving by the Ricci flow all obey parabolic equations with quadratic
nonlinearities. Knowing this, anyone familiar with the smoothing proper-
ties of parabolic equations would expect that appropriate bounds on the
geometry of a given Riemannian manifold (Mn, go) would induce a priori
bounds on the geometry of the unique solution g (t) of the Ricci flow such
that g (0) =go. Moreover, one would even expect the geometry to improve,
at least for a short time.
In this chapter, we verify those expectations by proving global short-
time derivative estimates for all derivatives of the curvature of a solution
to the Ricci flow. As we remarked in Chapter 6, we call these Bernstein-
Bando-Shi estimates (briefly, BBS estimates) because they follow Bern-
stein's technique of using the maximum principle to establish gradient esti-
mates, and were applied to the Ricci flow in papers authored independently
by Bando and Shi.
- Global estimates and their consequences
Local derivative estimates are important for performing dimension re-
duction and in other cases where one wants to take a 'semi-global limit' of
a sequence of geodesic balls whose radii go to infinity, or a 'local limit' of
a sequence of geodesic balls of uniform size. Local derivative estimates will
be discussed in the successor to this volume. In the present chapter, we will
prove those global derivative estimates that are important for establishing
long-time existence of the flow, as was discussed in Section 7 of Chapter 6.
Our main goal for this chapter is to prove the following result.THEOREM 7.1. Let (Mn, g (t)) be a solution of the Ricci flow for whichthe maximum principle holds. (This is true in particular if Mn is compact.)
Then for each a > 0 and every m E N , there exists a constant Cm dependingonly on m, and n, and max{a, 1} such that if
a
!Rm (x, t)lg(x,t) ::; K for all x E Mn and t E [O, K],223