1547671870-The_Ricci_Flow__Chow

100 4. MAXIMUM PRINCIPLES

Extend v to a vector field V defined in a space-time neighborhood of (xi, ti)
by the method described above. Then ( 4.10) and the null-eigenvector as-
sumption imply that at (xi, ti), we have

o ~ at a ( (Ac:)ij viv1. .) = ( at a A'° ) ij viv1..

[ (.6..A.o)ij + f3ij (A.o, g, t)] ViVj

= .6 ((A.o)ij vjvj) + f3ij (A,o,g, t) vivj ~ o.

This contradiction proves the claim. Then since o > 0 depends only on

MT;[~,T] I :t^91

and K, and is in particular independent of c, we can let c "'>. 0. Theorem

4.6 follows. D

REMARK 4. 7. The proof above corrects a minor oversight in the original

argument of Section 9 of [58], which failed to consider the ~ term.

3. Advanced weak maximum principles for systems
   Theorem 4.6 was proved in [58]; it may be regarded as the tensor ana-
   logue of Theorem 4.2. There is a tensor analogue of Theorem 4.4 which
   shows how a tensor evolving by a nonlinear PDE may be controlled by a
   system of ODE; it was proved in [ 59 ]. We shall state the result here but
   postpone our discussion of its proof.
   The set-up is as follows. Let Mn be a closed oriented manifold equipped
   with a smooth I -parameter family of metrics g (t) : t E [O, T ] and their Levi-
   Civita connections V' (t). Let 7f : £ --+Mn be a vector bundle over Mn with
   a fixed bundle metric h. Let

V (t) : C^00 (£) --+ C^00 (£ ® T* Mn)

be a smooth family of connections compatible with h in the sense that for

all vector fields X E TMn, sections r.p, 7/J E C^00 (£), and times t E [O, T ] one

has

X(h(r.p,7/J)) = h(Vxr.p,7/J) +h(r.p,Vx7/J).

Define

~ (t): C^00 (£ ® T Mn)--+ C^00 (£ @T Mn® T* Mn)

for all X E T Mn, ~ E T* Mn, and r.p E C^00 ( £) by

~ X ( r.p@ ~) ~ V xr.p@ ~ + r.p@ \i' X~·

Then the time-dependent bundle Laplacian 6. ( t) is defined for all cp E

C^00 ( £) as the metric trace