(2) Bando's result that solutions to the Ricci flow on closed manifolds
are real analytic in the space variables for positive time. The proof of this is
based on keeping track of the constants in the higher derivatives of curvature
estimates and summing these estimates.
Chapter 14. In Chapter 7 of Volume One we encountered the global
derivative of curvature estimates. The idea is to assume a curvature bound
K for the Riemann curvature tensor and, by applying the weak maximum
principle to the appropriate quantities, obtain bounds for the ra-th deriva-
tives of the curvatures of the form | V
In this chapter, we
present Shi's local derivative of curvature estimates. The idea of localizing
the derivative estimates is simply to multiply the quantities considered by
a cutoff function.
For the global first derivative of curvature estimate we previously con-
sidered t \ V Rm| + C |Rm| . This quantity does not seem to adapt well to
localization. Instead we consider rjt
+ |Rm| j |VRm| , where rj is a
cutoff function. The local first derivative estimate we prove says that if a
solution is defined on a ball of radius r and time interval [0, r] and if it has
curvatures bounded by K, then we have
CK(\ + - + K
\rz r
on the concentric ball of radius r/2 on the time interval [r/2, r].
Local higher derivative estimates are proven using a similar idea. For
example, to bound the second derivatives, one applies the weak maximum
principle to the quantity
Rm| , where the constant
A is chosen appropriately. We also discuss a version of the local derivative
estimates where bounds on the higher derivatives of the curvatures of the
initial metric are assumed up to some order. In this case we obtain improved
bounds for all higher derivatives of the curvatures. This result is useful in
an approach toward constructing Perelman's standard solution.
For complete solutions with bounded nonnegative curvature operator,
the local derivative estimates, when combined with Hamilton's trace Har-
nack estimate, yield instantaneous local derivative bounds. Previously, in
Chapters 7 and 8 of Part I of this volume, we saw applications of the local
derivative estimates to the study of the reduced distance and the reduced
We also briefly discuss D. Yang's local Ricci flow. Here the velocity
—2 Re of the metric is multiplied by a nonnegative weight function with
compact support. The local Ricci flow provides another approach to Shi's
short time existence theorem for the Ricci flow on noncompact manifolds.
Chapter 15. This chapter discusses differential Harnack inequalities
of Li-Yau-Hamilton-type for the Ricci flow. These gradient-type estimates,
which are directly motivated by considering quantities which vanish on the
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