1547671870-The_Ricci_Flow__Chow

98 4. MAXIMUM PRINCIPLES

and then taking V to be independent of time. To see that the Laplacian of V
vanishes at (xo, to), choose any frame { ei E Tx 0 Mn }7= 1 which is orthonormal
with respect tog (to) and parallel translate it in a spatial neighborhood along
geodesic rays emanating from x 0. With respect to this local orthonormal
frame, the Laplacian of V is
n
6V(xo,to) = L ['Vei (\7eiV)- \7V'eieiv] (xo,to)
i=l
n
= L ['VeiO- \7 0 V] (xo, to)= 0.
i=l
Then at any point in the space-time neighborhood of (xo, to), one has

Now since (o:ijViVj) (x 0 ,t 0 ) = 0 and (o:ijViVj) (x,t 0 )?: 0 for all x in a

spatial neighborhood of xo, one has

6 ( o:ij vivj) ?: o.

But equations (4.6) imply that at (xo, to),

6 (o:ijvivj) = (6o:ij ) vivj.

Combining these observations with the assumption

(f3ij vivj) (xo, to) ?: 0

shows that

~ at (ai] · .vivj) = 6 (a i] .vivj) + /Ji] iQ .vivj > _ o

at (xo, to). Hence, if O:ij ViVj ever becomes zero, it cannot decrease further.

Now we give the rest of the argument.

PROOF OF THEOREM 4.6. Given any T E (0, T), we shall show that

there exists 15 E (0, T] such that for all to E [O, T - 15 ], if o:?: 0 at t =to, then

o:?: 0 on M x [to, to+ 15]. The theorem follows easily from this statement.

F ix any to E [O, T - 15]. For 0 < c ~ 1, consider the modified (2, 0)-tensor

Ac defined for all x E Mn and t E [to, to+ 15] by

Ac ( x, t) ~ o: ( x, t) + c[ 15 + ( t - to)] · g ( x, t) ,

where 15 > 0 will be chosen below. As in the proof of the scalar maximum

principle, the term cl5g makes Ac strictly positive definite at t = to because

Ac (x, to)= o: (x, to)+ cl5g (x, to) > 0,

and the term c ( t - to) g will make

a a

at AC> at a