268 34. CMC SURFACES AND HARMONIC MAPS BY IFT
on 8 M. In particular, since forker (£ 9 ) = 0, we have that the formal cokernel is
trivial: (forimage (£ 9 ))1-= 0.
3.3. Weak solutions of the linearized equation and regularity.
Let (Mn,g) be a compact Riemannian manifold with bounda ry 8 M. By the
trace theorem (see Taylor [400]),^2 there exists a bounded linear operator, called
a trace operator,
T : W^1 '^2 (TM) -t w^1 /^2 ,^2 (TM laM)
such that T (U) = U laM for U continuous on M (more generally, we have that T:
Wk,^2 (TM) -t wk-^1 /^2 ,^2 (T MlaM) is bounded for k > 1 /2). See §2 of Appendix
K for the definitions of these spaces.
We also have the boundary tangent and normal projection maps
T: w112,2 (T MlaM) -t w1;2,2 (T (8M)),
j_: wl/2,2 (T MlaM) -t wl/2,2 (N (8M)),
respectively, where N (8M) -t 8 M is the normal line bundle. In particular , we
have the composite maps To T and 1-o T.
Define t he Banach space
W.}'^2 (TM)~ {U E W^1 '^2 (TM): T(Uh = o}.
Motivated by (34.34) regarding the linearization of q>, define the bilinear form
I : W.}'
2
(TM) x W.}'
2
(TM) -t JRby
(34.40) I (U, V) ~ ( ( (V'U, V'V) -Rc(U, V)) dμ - ( II(U, V)dCJ,
JM JaMwhere II(U, V) ~ II(T(U), T(V)). By the trace theorem, I is bounded.
Now assume that g has negative Ricci curvature and 8 M is concave. Then
there exists a constant 5 > 0 such that we have the coercivity estimate
(34.41)
for U E W.}'^2 (TM).
In Theorem K.1, take A= B = W-}-'^2 (TM) and f =I. Then , by (34.41), we
h ave
LEMMA 34.14 (Weak solutions of the linearized equ ation). Let (Mn, g) be acompact manifold with negative Ricci curvature and concave boundary 8M. For
any Q E £^2 (TM) and any f E L^2 (T (8M)), there exists a unique U E W-}-'^2 (TM)
such that for all VE W-}-'^2 (TM) we have
(34.42) I (U, V) = r (Q, V) dμ + r (!, V) dCJ.
JM JaM
(^2) In Proposition 4 .5 on p. 287 of [400] the trace theorem is stated for functions. One may
extend this to section s of vector bundles by locally trivializing the bundles.