1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. HARMONIC MAPS AND THEIR LINEARIZATION

In view of this, we compute that

("V/vjdf)~ - ("Vj"Vkdf)~ = (Vi"Vjdf)~ - ("Vj"Vidf)~

= _ (RM)e 8f

8

+ (RN)8 8!°' 8ff3 8j'Y

ijk axe a.f3'Y 8xi oxJ 8xk.

Tracing this by multiplying by gjk, we obtain

from which (K.39) follows.

and

Define

( M) (( M) ) ( M)em of

8

Re (df 0 di)~ Re (di), df g!Zlh =ho" Re axe oxm ojf'

(RN) (df 0 df 0 df 0 di) ~ (tr^2 '^3 9 ((RN) (df, di) df), df) g!Zlh

= imh jk (RN)& 8!°' 8ff3 oj'Y ojf'

g^8 "^9 a.f3'Y 8xi f)xJ 8xk oxm.

By (K.39) and (K.38), we have the following.

345

LEMMA K.18 (Evolution of the energy density). Under the harmonic map heat

flow we have

(K.40)

8

8

ldfl! 0 1t = t:. 9 ldfl! 0 h - 2 [v^90 h (df)[

2

• 2 (RcM) (df 0 di)
  t g0h

• 2 (RN) ( df 0 df 0 df 0 di).

In particular, if the sectional curvatures of (N, h) are nonpositive and the Ricci

curvatures of (M,g) are bounded below by-KE JR, then

(K.41)

Assuming that M is closed, we may apply the maximum principle to (K.41)

and conclude that jdfl! 0 h (x , t) is bounded above by Ce^2 Kt on M x [O, oo), where

C = max ldf l!01t (" 0).

Second, we shall show that the norm squared of the Laplacian of f is a subso-

1 ution to the heat equation. By taking V = t:. 9 ,hf in (K.33), we see that

gt (t:.g,hf)'Y = (t:.g,h (t:.g,hf))'Y - (t:.g,hf)f3 r (h)~f3 (t:. 9 ,hft

8 N 'Y i 8!°' off3

+ (t:.g,hf) (R )fof3g J f)xi 8xJ