1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. HARMONIC MAPS AND THEIR LINEARIZATION 343

we may rewrite (K.34) as

(K.35) :s/s=O (V,b.g,hfs)J;h = h<>1,V"' (b.g,hV),, + (RN)fo/3c vo: gij; V"

• h"',, V^13 (Vf hV)"' (6 9 ,hf)l'.

When V = Vs depends on s, we simply add the term ( ~ ls=O, 69 ,hf) f•h to the

RHS of (K.35).

In conclusion, we have the following.

LEMMA K.15 (Second variation formula for map energy). If~ ~ Vs and

Vo= V , then

where

tr 9 ( (RN) (V, df, df, V)) ~ (RN) fo/3c V^0 ~~: gij ~~; V".

Assume that V has compact support in the interior of M. Then differentiating

(K.29) and integrating by parts yields

In particular, if f is a harmonic map, then

(K.36) :: 2 /s=O E 9 ,h Us)= 2 JM (f vf*hvr:f81h - t r 9 ((RN) (V, df, df, V))) dμ 9

~ I(V, V).

The RHS of (K.36) is called the index form.
We say that a harmonic map f is weakly stable if I (V, V) ~ 0 for all V E
f* (TN) with compact support in the interior of M. If (N, h) has non positive
sectional curvature, then

I (V, V) ~ 2 { [vf*hvf 2 dμ 9 ~ 0 ,

jM gf81h

so that any harmonic map f mapping into a Riemannian manifold with nonpositive

sectional curvature is weakly stable.

EXERCISE K.16. Adopt the notation in (K.30). Show that

\l a;as (b.;:1hF) = 69 ,h V +tr 9 (RN (V, df) df).