- HA R M O N IC MAPS AND THEIR LINEARIZATION 341
3 .3. Second variation formula for the map energy.
We adopt t he notation of t he previous subsection. In particular, let V ~~ ls=O. Although in local coordinates the scalar ts ls=O (6.. 9 ,h f s )'Y is well de-
fined , the expression ts ls=O (6.. 9 ,hfs ) is not well defined since 6.. 9 ,hfs is a section
of J; (TN) which is a bundle that dep ends on s. With this understanding, by
differentiating (K.20a) we see t hat t he first variation of the m ap-Laplacian is given
by(K.32) -a a I ( UA g h f S )'Y _ - UA g ( v 'Y) + g ij (r ( h )"I a(3 0 1 ) -a av"'. -a a Jf3.
S s=O , xi xJij (r (h)'Y of) a!°' avf3
+ g a(3 axi 8 xJi. ( ( o a 'Y ) ) a r a Jf3
+ g J v 8 y 6 r ( h) af3 0 f 8 xi 8 xJ.
We wish to put this formula in a nicer form. Define
From now on we shall sup press the compositions with f in our formulas. We have
and
Hence