- HARMONIC MAPS NEAR THE IDENTITY OF MANIFOLDS WITH Re < O 267
3.2. The linearization of <!? and the function spaces for the IFT ar-
gument.
We first assume that all quantities, such as maps and sections of vector bundles,
are smooth enough to obtain the formulas. By (K.43) and (K.44) in Appendix K,the linearization of<!? (g, F) at (g, id) and with respect to the second variable F is
given by
(34.33) ((D2)(g,id) <!?) (V) = (D.gV + Rc(V), ('VNV)T -II(V)) E 2(M) x2(8M)for vector fields Von M with VJ_ = 0 on 8M.
Note that for vector fields V and Won M with VJ= 0 and WJ = 0 on 8M,
we have by integrating by parts that(34.34) ( (D.g V + Rc(V)) · W dμ - ( ((\7 N V)T - II(V)) · W dcr
JM JaM= - ( ('VV, 'VW) dμ + ( II(V, W)dcr + ( Rc(V, W)dμ,
JM JaM JM
where we have used (\7 N V)T · W = (\7 NV) · W and where dcr is the induced volume
form on 8M. Hence, if VE ker((D2)(g,id) <!?),then by taking W =Vin the above
equation, we have(34.35) - ( IVVl^2 dμ + ( II(V, V)dcr + ( Rc(V, V)dμ = 0.
JM JaM JM
Next, we define the function spaces for the IFT argument that we shall apply tothe map <!?. Let Ck,a (S!T* M) denote the Banach manifold of Ck,a Riemannian
metrics on M. Let Ck·°'(M; 8M) denote the Banach manifold of Ck,a diffeomor-
phisms of (M, 8M) to itself. Let C~·°' (TM) denote the Banach space of Ck,a
vector fields U on M with UJ_ = 0 on 8M. Let Ck•°'(T(8M)) denote the Banach
space of Ck,a vector fields on 8M. (See §2 of Appendix K for more details on the
definitions of these spaces.) Consider the map(34.36) <!?: C^2 '°' (S!T* M) x C^2 '°'(M; 8M) ~ C°' (TM) x C^1 '°'(T(8M))
defined by (34.31). The linearization of<!? at (g, id) and with respect to the second
variable F,
(34.37)
is given by (34.33).Now assume that Re< 0 on M and II (8M) :S 0. In the next two subsections
we shall show that (D 2 )(g,id)<I? in (34.37) is an invertible operator (see Lemma 34. 17
below). To motivative this, we observe some formal calculations.First, if V satisfies (34.35), then V = 0. That is , the formal kernel
(34.38) forker((D2)(g,id) <!?) = 0
is trivial.
Second, by (34.34) we have the following formal self-adjointness property for
the operator Lg (V) = Dog V + Re (V):
(34.39) (Lg (V) ' W)L2(TM) = (V,Lg (W))L2(TM)
provided ('VNV)T - II(V) = 0 and ('VNW)T - II(W) = 0 on 8M.
By (34.39), we have that V E forker (Lg) and (\7 N V)T - II(V) = 0 on 8M
if and only if (V,Lg (W))L2(TM) = 0 for all W satisfying ('VNWh - II(W) = 0