1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

32 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

(see Theorem A.57 in Part I for the corresponding linear trace Harnack

estimate). When X = \i' f, we may rewrite this as

(17.92) L (v, \i'f) = (div-l\7/) o (div-lV'f) v + (Rc+\i'\i' f, v)

(17.93) = ef (divodiv+Rc+\i'\i'f) (e-fv).

Observe also that

8R

L (2Rc,X) = 8t - 2 (\i'R, X) + 2Rc (X, X)

is Hamilton's trace Harnack quadratic (see (15.17) in Part II).

Let V = gijVij·Two of the above quantities are related by the following

(see also Lemma 6.82 in Part I)

LEMMA 17.29 (Variation of Perelman's modified scalar curvature). If

g 8 g = v and g 8 f = V (so that gs (e-f dμ) = 0), then

(17.94) :s ( R + 2~f - l'\7 fl^2 ) = L (v, \i' f) - 2 (v, Re +\i'\i' !).

Integrating the above formula by parts, we obtain (see §1.1 of [152];

compare with Exercise 6.16 in Part I)

LEMMA 17.30 (Perelman's first variation formula for F). If gsg = v and

g 8 f = V, then

(17.95)

(17.96)

! Fi (g, f) = -JM (v, Re +\i'\i' !) e-f dμ

=- ( L(v,\i'f)e-fdμ.

}M.

4.2. Some 2-loop formulas.

Let

(~Lh)ij ~ ~hij + 2Rkij£.hkc - Rikhkj - Rjkhki

be the Lichnerowicz Laplacian, which acts on symmetric 2-tensors. Given

a function f on M, define the symmetric 2-tensor H\i'f, which is a form of

Hamilton's matrix Harnack quadratic, by (compare with the expres-
sion in (15.11) of Part II)

(17.97)

H'Vf (X, Y) ~ ( ~LRc-t\i'\i'R + Rc^2 ) (X, Y)

+ P (X, \i' f, Y) + P (Y, \i' f, X) +Rm (\i' f, X, Y, \i' J),

where
P (X, Y, Z) ~ (\i' x Re) (Y, Z) - (\i'y Re) (X, Z)

for tangent vectors X, Y, Z (see Chapter 15 in Part II for the proof of Hamil-
ton's matrix Harnack estimate). This may be rewritten as

(17.98) H\i'f = ef (divodiv+Rc+V'V'fh, 4 (e-fRm),