1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

194 5. ENERGY, MONOTONICITY, AND BREATHERS

The first quantity vanishes on steady gradient solitons fl.owing along \7 f,

whereas the second appeared in (5.4).^5 We call RiJ and Rm the modified

Ricci curvature and modified scalar curvature, respectively; they are

natural quantities from the perspective of the Ricci flow. We can rewrite

F (g, f) =JM gij Rije-f dμ =JM Rme-f dμ

and

when V = 2h.

1.3. The modified Ricci and scalar curvatures. In this subsection

we digress by showing RiJ and Rm are natural quantities. Consider a closed

Riemannian manifold (Mn,g) and a metric g = e-~f g conformal tog. Let

Rij =Re (g)ij, Rij =Re (g)ij, R = R (g), and R = R (g). The Ricci and

scalar curvatures are related by (see for example subsection 7.2 of Chapter

1 in [111] or (A.2) and (A.3) in this volume)

(5.16)

Rij - = Rij + ( 1--2) \7i\7jf + -b..fgij^1 + n --^2 n-^2 2

n n n^2 -\7d\7jf--n^2 - l\7fl 9ij·

Tracing this yields

(5.17) R = e~f (R+ 2 (n-1) b..f-(n-1) (n-2) l\7fl2).

n n2

The volume forms are related by dμ 9 = e-f dμ and the total scalar curvature

of g is given by

JM Rdμg =JM e-n;2 f ( R+ (n - l~~n - 2) l\7fl2) dμ,

where we integrated by parts, i.e., we used

r e-n;2 f b..f dμ = n - 2 r e-n;2 f l\7f12 dμ.

JM n JM

Now consider the Riemannian product (Mn; g) x (Tq, hg), where (Tq, hq)

is a flat unit volume q-dimensional torus. The formulas for the Ricci cur-

vature and scalar curvature of metric e-n!qf (g + hg) are given by (5.16)
and (5.17), respectively, where we replace n by n + q. If we take the limit
as q __, oo while fixing (Mn,g), then we obtain Perelman's modified Ricci
tensor:

(5.18) q-+oo lim Re (e-n!qf (g + hg)) =Re +\7\7 f

and Perelman's modified scalar curvature:

(5.19) q-+oo lim R (e -n!qf (g + hq)) = R + 2b..f - l\7 fl^2 ,

(^5) Earlier we also encountered these quantities in Chapter 1.