- HARMONIC MAPS NEAR THE IDENTITY OF sn 261
2.1. Linearization and its kernel of the map-Laplacian on sn.
Let g and g be C^00 Riemannian metrics on a manifold Mn and consider adiffeomorphism F: (M, g) ---+ (M, g). Its map-Laplacian 6..g, 9 F, defined by (K.19),
is a section of the vector bundle F* (TM) ---+ M. To remove the dependence of
this bundle on F, we pull back the map-Laplacian to get a vector field on M:
(34.8)
In local coordinates, the map-Laplacian is given by(34.9) (6..g 9Ft = 6..g (Fa)+ gij (f'bc 0 F) ~Fb ~Fe'
' v~v~
where 6..g denotes the Laplacian, with respect tog, acting on functions, and where
the f'bc denote the Christoffel symbols of g. Hence, its pull-back is(34.10) ( (F-'). (L>,,;F))' ~ (a(::.')' o F) L>, (F")
.. (- ) (8 (F-
1)k ) 8 F b 8Fc
- g'J rbc o F oya o F 8xi oxJ.
The linearization Lg : C^00 (T M) ---+ C^00 (T M) at t he identity map of the oper-
ator F H (F-^1 L (6..g,gF) is given by
(34.11) L g (V) ~ 6..g V +Reg (V),where 6..g is the rough Laplacian (see (K.43) in Appendix K). We next investigate
the obstacles to inverting Lgsph for the unit n -sphere ( sn, 9sph).
Let 6..d ~ - (do + od) denote t he Hodge-de Rham Laplacian acting on
differential forms. We say that A is an eigenvalue of 6..d with eigenform a if
(6..d +.A) a= 0.^1 Let >-l^1 l (Sn), k 2 0, denote the k-th eigenvalue of 6..d acting on
1-forms on (Sn, 9sph)- By identifying vector fields with their metrically dual 1-forms
on (Sn,9sph), we see that the eigenvalues of the linearization Lgsph: C^00 (TSn)---+
C^00 (T sn) satisfy
(34.12)
Ak(Lgsph) = Ak(6..gsph + Rcg.,,h) = >-l^1 l (6..d + 2 (n - 1)) =A~] (Sn) - 2 (n - 1),
where we used that Rcg.,,h = (n - 1) 9sph and t hat 6..ga = 6..da +Re (a) for any
1-form a.
Let c.>-l^1 l (Sn) and cc>-l^1 l (Sn) denote the k-th eigenvalue of 6..d acting on closed
and co-closed 1-forms on (Sn,9sph), respectively. Regarding the RHS of (34.12), by
taking p = 1 in (K.54), we have that
(34.13) c >-l^1 l (Sn)= (k + 1) (n + k),so that the two lowest eigenvalues of 6..d acting on closed 1-forms are
(34.14) c>-b^1 l(sn)=n and c.A~^1 l(sn)=2(n+l).
On the other hand, taking p = 1 in (K.55), we have that(34.15)(^1) For eigenvalues of the Laplacian, the geometer's convention is opposite that of the analyst's.