1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

494 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

Hence, if we take ei ® ei, ei ® e2 + e2 ® ei, ei ® e3 + e3 ® ei, e2 ® e2, e3 ®.

e3, e2 ® e3 + e3 ® e2 as a basis for S^2 T* M, then

er DX (g) (()

-2E^1 mVlm + EmCVR.m + E^11 vn

-E1Cvc2 + E^11 v12

• EuVR.3 + E^11 v13
  E11v22
  E^11 v33
  E^11 v23
  E^22 v22 + E^33 v33 + 2E^23 v23
  -E12V22 - E13V23
  -E12V23 - E13V33
  Env22
  E^11 v33
  Env23

The symbol as a matrix is given by

0 0 0 E22 E33 2E23

0 0 0 -E12 0 -E13

er DX (g) (() =

0 0 0 0 -E13 -E12

0 0 0 En 0 0

0 0 0 0 En 0

0 0 0 0 0 En

Since this matrix is upper triangular, its eigenvalues are 0 and E^11 , which

are nonnegative.

To eliminate the degeneracy, we apply DeTurck's trick. As in (3.29) on

p. 80 of Volume One, given a fixed torsion-free connection f', define the

vector field w = w (g, r) by

Wk, -;-g pq (rk pq _ r-k) pq.

Consider the second-order operator

Y (g) ~ Cwg.

We have