1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

340 K. IMPLICIT FUNCTION THEOREM

To obtain the third equality above we used the divergence theorem and

(K.27) div^9 ((V, df) 1 .h) = ('vr1iv, df)

90

h + \v, tr 9 ( V^90 hdf)) J•h,

where ( (V, df) f *h)i = (haf3 of) V°' (dft

Summarizing, we have the following.

LEMMA K.14 (First variation formula for the map energy). If ofas. I s=O ==. V '

then

(K.28) ~^81 ldfslg0h^2 - - 2 \ \7 j*h V, df ).

us s=O g0h

If V has compact support in th e interior of M, then

(K.29) : I Eg,h Us) = -2 r (V, b.g,hf) f *h dμg.

S s=O JM

Hence, a map is a critical point of the map energy if and only if it is a harmonic

map.

Next we give an alternate, invariant, proof of (K.29). Define

(K.30) F: M x (-10, io)--+ N

by F(x, s) ~ fs(x). We have DF E C^00 (T(M x (-10,io))®FTN). Abusing

notation, let T* M also denote the pull-back by the projection M x (-10, io) --+ M

of T* M. Let dMF E C^00 (T* M 0 F*TN) denote the spatial derivative of F , let

%s denote the tangent vector in the positive ( -10, c) direction, and let V denote the

natural connection on T M 0 FTN induced by g , h, and F.

For any X E TpM, let X = X E C^00 (T(M x (-c,c))l{p}x(-e:,e:)). Since

V 8/asX = 0 and V (dF) is symmetric, we have

(K.31)

Using this, one computes

: I Eg,h Us)= 2 r : I (dMF, dMF)g0h dμg

S s=O JM S s=O

= 2 JM (Va/as (dMF)ls=O, df) grslh dμ 9

= 2 JM \vf*h (~~) , df)grslh dμ 9 ,

from which one deduces (K.29) by integration by parts, i.e., (K.27), and ~~ = V.