1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. THE CROSS CURVATURE FLOW 493

THEOREM B.21 (XCF short-time existence). If (M^3 , g 0 ) is a closed 3-

manifold with either negative sectional curvature everywhere or positive sec-

tional curvature everywhere, then a solution g (t), t E [O, c:), to the cross

curvature flow with g ( 0) = go exists for a short time.

PROOF. We only consider the case of negative sectional curvature and

leave the case of positive sectional curvature as an exercise for the reader.

By Remark 3.4 on p. 69 of Volume One, if g 8 gij = Vij is a variation of the

metric gij, then

a 1

at Rijkf. = 2 (\7 /v eVjk + \1 j \1 kVif. - \1 i \1 kVje - \1 j \1 f.Vik)

(B.34)

1

+

2

gpq (Rijkpvqe + Rijp.evqk),

so that

where the dots denote terms with 1 or fewer derivatives of v. Applying
(B.31) to (B.30), we obtain

Thus the linearization of the map X which takes g to 2c is a second-order
partial differential operator. Its symbol is obtained from 2 gs Cij by replacing
a~i by a cotangent vector (i in the second-order terms:

here DX (g) denotes the linearization of X at g. Since the sectional curvature

is negative, Eme is positive, which in turn implies that the eigenvalues of
the symbol are nonnegative.

In analyzing the symbol aDX (g) ((),without loss of generality we may

assume (i = 0 and (2 = (3 = 0. Then

[O"DX (g) (() vLj = -8i1Eievej - Oj1Em^1 vim + 8i18j1Emev.em + E^11 vij·