1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

190 D. SALAMON, FLOER HOMOLOGY

then u" has a convergent subsequence. More precisely, the subsequence can be

chosen to converge uniformly with all derivatives on compact sets. The limit curve

is again a J-holomorphic curve and, if E(u") :S c for all v, then the energy of

the limit curve is also bounded by c. This result extends to the case where u"

is a sequence of J"-holomorphic curves and J" converges in the c=-topology to

J E :J(M,w). The proof follows from standard elliptic bootstrapping techniques.

For details see [31]. D

Let us now examine the case of a sequence of J-holomorphic curves u" : C ____, M
with uniformly bounded energy but unbounded derivatives:

Such sequences do exist. Note, in particular, that the condition sup" E( u") < oo

is equivalent to a uniform L^2 -bound on the first derivatives of u". In contrast, a
uniform LP-bound on the first derivatives for some p > 2 would imply a uniform£= -
bound and hence, by Remark 4.1 (iii), the existence of a convergent subsequence.
That such bounds cannot be obtained in the case p = 2 has an analytical and a
geometric reason. The analytical reason is the fact that p = 2 is a borderline case
for the Sobolev estimates (see Section 2.3). The geometric reason is the conformal
invariance of the energy and the resulting bubbling phenomenon, which was first
observed by Sacks and Uhlenbeck in the context of harmonic maps [44]. Here is
how this works.
Suppose that z" E C is a sequence such that

V->lim CXl ldu"(z")I = oo, lim z" = z.

1/->CXl

Modifying the sequence slightly, without changing its limit, we may assume that

there exists a sequence 8" > 0 such that

V->lim CXl 8" = 0, lim 8"ldu"(z")I = oo,

V->CXl

sup ldu"I :S 2ldu"(z")I.

B6v (zv)

(See the footnote on page 174.) Now consider the rescaled sequence

v"(z) = u" (z" + c:"z), c:" =

1

ldu"(z")I.

This sequence has uniformly bounded derivatives on any compact subset of C.

Namely, for any R > 0 there exists a VR such that Re" :S 8" for v ;::: VR. Hence

ldv"(z)I :S 2 for lz l :S Rand v;::: VR· By Remark 4.1 (iii), this implies that there

exists a subsequence which converges, uniformly with all derivatives on compact
subsets of C, to a J-holomorphic curve v : C ____, M. This curve has finite energy
and so, by Remark 4.1 (i), extends to a J-holomorphic sphere. Since ldv(O) I =

limv_,= ldv"(O)I = 1, the map vis nonconstant. Hence, by Remark 4.1 (ii), it has

energy
E(v) ;::: n.

Now the energy of v" in BR(O) is equal to the energy of u" in BRE:v(z") which in
turn is bounded above by the energy of u" in an arbitrarily small ball Bf;(z ) for
v sufficiently large. This means that in the large v limit the sequence u" has an
energy of at least n concentrated in an arbitrarily small ball about z. Hence, as in
the proof of Proposition 3.3, we have

lim lim inf E( u", Bf;(z)) ;::: n.

c-+0 11 -+oo