LECTURE 2. ANALYTICAL TOOLS 57- Study the Sobolev embedding results, which are related to the W^1 ·P-norms
p > 2.
Assume next that M is a closed orientable three-manifold equipped with acontact form>. and J an admissible complex multiplication for~· Denote by J the
associated almost complex structure on ffi. x M.Theorem 2.3. Let g be a metric on M and assume c > 0 is a given constant.
Then given E > 0, there exists a sequence Ck > 0 such that for ii: D , --+ ffi. x M with
the fallowing holdsiis + ](u)iit = 0
u(O) E {O} x MIVii(z)I :::; c for z ED"
fork EN.
For the latter view M as lying in some ffi.n using Whitney's theorem.Proof. Fix k E N. Arguing indirectly we find a sequence ( iie) such that
(iie)s + i(ue)(ue)t = 0
iie(O) E {O} x Mfor z ED,
lliiellck(D,el --+ +oo
Ee~ 0.Using the Ascoli Arzela Theorem we may assume after taking a subsequence
in C^0 (D,).Taking a suitable chart around iio(O) we may assume for some 0 < E* :::; E
iie: D e --+ C^2](O) = i, j on C^2
l'Viie(z)I :::; c, z ED,.
lliiellwk'.P(D,e,c2)--+ oofor some k' > k + 1 (by the Sobolev embedding theorem). By Theorem 2.3 the
gradient bound implies Wk',P-bounds on some D,, giving a contradiction. 0
Let us mention the following. Assume we have a solution ii ofii: [O,oo) x 81 --+ ffi. x Miis + i(u)iit = 0.
View ii as a map periodic in the second variable, i.e.
ii : [O, oo) x ffi. --+ ffi. x M.By the preceeding results uniform gradient bounds on [O, oo) x ffi. imply uniform
Ck-bounds on [E, oo) x R