1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

300 35. STABILITY OF RICCI FLOW

If we now specify a compact E instein manifold (Mn,g) with Re = -rg for

some r > 0, then g is a fixed point of the rescaled Ricci fl.ow (35.21). After ap-

plying the usual DeTurck trick, we verify that the linearization of this flow at the

fixed point g is given by the formally self-adjoint operator L that sends a smooth

symmetric (2, 0)-tensor h to !:::.Lh - 2rh, where !:::.L is the Lichnerowicz Laplacian.

In coordinates, one has

(Lh)ij = (!:::.hij + 2Ripqjhpq - R7hkj - Rjhik) - 2rhij

= !:::.hij + 2Ripqj hpq,

where (as usual) all indices are raised using g and covariant differentiation is carried

out using the Levi-Civita connection of g. Define a smooth, compactly supported

function Q(h) by

Integration by parts gives

(35.22) (Lh, h) = - JJVhJJ^2 + 2 { Q(h) dμ.

}Mn

Define a (3, 0)-tensor T(h) by Tijk = \i'khij - \i'ihjk· Then Koiso's Bochner-type

formula [167]

IJVhJl^2 = JJbhJJ^2 + ~ llT(h)Jl^2 + { { (S(h), h) - R7hjkhij} dμ

2 }Mn

shows that

IJ\7hJJ

2

= ~ llTll

2

+ IJbhJJ

2

+ r JihJJ

2

+ { Q(h) dμ.

2 }Mn

This allows us to write (35.22) as

(35.23) (Lh, h) = -~ llTJl^2 - JJbhJJ^2 - r Jlhil^2 + { Q(h) dμ.

2 }Mn

One can refine this further by using the irreducible decomposition of the Riemann
curvature tensor Rm. Recall that for any n-dimensional Riemannian manifold, one
has
1 - 1 0 -
Rm = 2 ( ) R(g A g) + --
2
(Re A g) + W,
nn-1 n -
0
where A denotes the Kulkarni- Nomizu product of symmetric tensors, Re denotes
the trace-free Ricci tensor, and W is the Weyl tensor. On the Einstein manifold
(Mn, g), this reduces to

Rm = - 2 ( n r_

1

) (g A g) + W,

which one may write in local coordinates as
r
Rijke = ---
1
(gifgjk - 9ik9je) + wijke·
n -

If p, q are (2, 0)-tensor fields, define

Then one can write Q(h) as

W(p, q) = WijktP if q^1 'k.

Q(h) = r{lhl^2 - (tr 9 h)^2 } + W(h, h)

n-1