LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 195every complex automorphism of the 2-sphere has the form of a fractional linear
transformation (also called Mobius transformation)(z ) = az + b ad - be = 1.
<p cz+d'
Here the numbers a, b, c, d form a complex 2 x 2-matrix with determinant 1. Sincethe Mo bi us transformations associated to (a, b, c, d) and (-a, -b, -c, -d) are equal,
the group of Mobius transformations can be identified with the group
G = PSL(2,C) = SL(2,C)/{±Il}.Definition 4.4 (Equivalence). Two stable J-holomorphic curves
(u,z) = ({ua}aET, {za.e}aE/31 {ai,zi}1:::;i9),
(ii, z) = ( { ua } aET' { Zaf3} aE/3' { ai, zi}1:::;i9)of genus zero in M with k marked points are called equivalent if there exists a tree
isomorphism f : T -t T and a collection of Mobius transformations <p = { <fJa}aET
such that the following holds. ·
(i) For all a ET, UJ(a) = Ua o<p" --^1.(ii) For all a ,/3 ET with aE/3, ZJ(a)f(/3) = <fJa(za13).
(iii) For i = 1, ... , k, ai = f(ai ) and zi = <fJa, (zi)·
Definition 4.5 (Gromov convergence). A sequence
(UV, z") = ( { U~ }aETv, { Z~/3 }a Ev /31 { ai, zn1:::;i:::;k)
of stable J-holomorphic curves with k marked points is said to Gromov convergeto a stable J-holomorphic curve (u,z) = ({ua}aET,{za.e}ae.e,{ai , zi}1:::;i:::;k) if, for
v sufficiently large, there exists a surjective tree homomorphism f " : T -t T" and
a collection of Mobius transformations { <p~}aET such that the following holds.(i) For every a E T the sequence ujv(a) o <p~ : 82 -t M converges to Ua,
uniformly with all derivatives on compact subsets of 82 - Za. Moreover, if
f3 E T such that aE f3, then
ma.e( u) = lim lim E fv (a) ( u", <p~(B 0 (Zaf3))).
c--+O v--+oo
(ii) Let a, f3 E T such that aE/3 and let Vj be some subsequence. Iff"i (a) = f"i (/3) for all j then ( <p:; )--^1 o <p~i converges to Zaf3, uniformly
on compact subsets of 82 - { Zf3a}. If f"i (a) i= f"i (/3) for all j then
Zaf3 =^1. lmj--><xi ( <pa "i)--1( Z j"i (l/j a)j"i (/3) ) ·(iii) For i = 1, ... , k , ai = j"(ai ) and Zi = lim,,__, 00 (cp~,)--^1 (zi).
The previous definition is somewhat complicated and it is useful to record its
meaning in the case where the trees T" all consist of single points. This means
that each u" : 82 -t M is a single J-holomorphic sphere equipped with k distinct
marked points z!, ... , zk. Such a sequence Gromov converges to a stable map
( u , z) = ({ ua}aET, { Zaf3 }aEf3, { a i , zi }i::=;i:::;k) iff there exist sequences cp~ E G such
that the following holds.
(i) For every a E T the sequence u" o cp~ : 82 -t M converges to Ua uniformly
with all derivatives on compact subsets of 82 - Za. Moreover, if f3 ET such
that aE/3, then
ma13(u) = lim lim E(u" o cp~, B 0 (za13)).
c--+O v--+oo