1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

490 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

matrix Harnack inequality for the heat equation (see Part II of this volume),

if ( pN, g) has parallel Ricci tensor and nonnegative sectional curvature, then

\7\7 log u + 2 (,j-t) g 2:: 0 and hence

.!!_ [(T -t)(N-n)/^2 { udμl :'S 0.

dt Jxt(M)

Equality holds if and only if

ii - (\71ogu)1-= 0

and

( \7\7 log u + 2 (Tl-t) g) 1-= 0.

EXERCISE B.15. Prove Hamilton's generalization of Huisken's mono-

tonicity formula.

2.3. Monotonicity for the harmonic map heat flow. Similar in-

equalities hold for the harmonic map heat flow and the Yang-Mills heat

flow. The original monotonicity formula for the harmonic map heat flow

was discovered by Struwe [342] and extended by Chen and Struwe [90] and

Hamilton [161]. The monotonicity formula for the Yang-Mills heat flow

appears in [184].

THEOREM B.16 (Struwe and Hamilton). Let (Mn, g) and (Nm, h) be

Riemannian manifolds where M is closed. If F : (M, g) ---7 (N, h) is a

solution to the harmonic map heat flow

8F

at = 1). 9 ,hF,

u: M ---7 .IR. is a positive solution to the backward heat equation ~~ = -L). 9 u,

and T ~ T-t, then

See the above references for applications of the monotonicity formula to
the size of the singular set of a solution.

3. The cross curvature fl.ow

In this section we present the details to results for the cross curvature
flow stated in subsection 4.2 of Chapter 3 of Volume One.