346 K. IMPLICIT FUNCTION THEOREM
so that
:t l6.g,hfl
2
= 2h-yc; (6.g,hft :t (6.g,hf)^7 + :t (h-yc; 0 f) (6g,hft (6g,hf)^7= 2h 7 c; (6. 9 ,hf)c; ( (6. 9 ,h (6 9 ,hf))^7 - (6. 9 ,hf/3 r (h)~ 13 (6.g,hf)°')
h (A f)c (A !)8 (RN)"! ij of°' 8J
13
+ 2 -ye; Ll.g,h Ll.g,h 8a.{3 g 8xi axJ- (6g,hf)
8
( ()~8 h-yc;) (6g,hft (6g,hf)^7
= 6. l6 9 ,hfl^2 - 2 IV' (6. 9 ,hf)l^2 + 2 (RN) (6. 9 ,hf, df, df, 6.g,hf),
where
(RN) (6. 9 ,hf, df, df, 6. 9 ,hf) ~ h-yc; ( 69 ,hft ( 6. 9 ,hf)
8
(RN) ;o:f3 lj ~~: ~~~.In particular, if the sectional curvatures of (N, h) are nonpositive, then(K.42) :t l6.g,hfl
2
:::; 6. l6.g,hfl
2- 2 IV' (6.g,hf)l
2
:::; 6. l6.g,hfl2
.We m ay apply the maximum principle to (K.42) and conclude that 16. 9 ,hfl is uni-
formly bounded on M x [O, oo).
3.5. Linearization of the harmonic map equation with normal bound-
ary condition.
In §3 of Chapter 34 , we considered an existence problem for harmonic maps sat-
isfying the normal boundary condition. In particular, adopting the notation in thatsection, let Mn be a manifold with boundary 8M and let F be a diffeomorphism
from (M, 8M) to itself. Correspondingly, we defined in (34.31) the map
<I!(g, F) ~ ((F-^1 )* (6. 9 , 9 F), (F-^1 )* (F* (N)h).
We now compute the second component of the linearization of the map <I! at (g, id),
where id denotes the identity map of M.
Note that, formally, the tangent space at id of the space of diffeomorphismspreserving 8M comprises the vector fields V on M such that ( Vl 8 M)l_ = 0. Let
Fs be a I-parameter family of maps with Fa= id and fs Js=O Fs ~VE C^00 (TM).Given p E M, choose coordinates {xi} satisfying r~j (p) = 0. Then at p we have
by (K.32) and dds ls=O (Fs-l L (W) = [V, W] that
:s ls=O ( (Fs-l) * (6. 9 , 9 Fs)) k = 6. 9 (Vk) + lj ( 0 ~e r7j) Ve.
We also have
k.. k
(6. 9 V) =g' JV'iV'j V
- - g ij ( axi 8 ( axJ 8 vk + rk j£ ve) -re ij V' e vk + r k i£ V' j ve)
= 6 (Vk) + 9 ij (~rk) v e
g OX' JRand