LECTURE 4
Gromov Compactness and Stable Maps
The purpose of this lecture is to discuss how Gromov compactness for J-holo-
morphic curves leads to Kontsevich's notion of stable maps and to describe a natural
topology on the space of stable maps. The last section deals with the Deligne-
Mumford compactification of the moduli space of Riemann surfaces of genus zero
with marked points.
4.1. Bubbling
Let (M,w) be a compact symplectic manifold of dimension 2n and J E .J(M,w)
be a compatible almost complex structure. Suppose that u : C __, M is a J-
holomorphic curve. With coordinates z = s + it on C this means that u satisfies
the PDE
au au
as + J(u) at = o.
The energy of u is defined as the integral
E(u) = fc lasul
2= fc u*w.
Throughout we shall only consider J-holomorphic curves with finite energy. In
the first section we shall discuss the limit behaviour of sequences with uniformly
bounded energy. The following three facts play a central role.
Remark 4.1. (i) The removable singularity theorem asserts that every J-ho-
lomorphic curve u : C __, M with finite energy extends to 82 = CU { oo }. This
means that the function C - {O} __, M: z f-+ u(l/z) extends to a smooth map on
C. A proof can be found in [31].
(ii) By Lemma 3.2, there exists, for every J E .J(M,w), a constant n =
n(M,w, J) > 0 such that
E(u) ~ n
for every J-holomorphic sphere u: 82 __, M.
(iii) If uv : C __, M is a sequence of J-holomorphic curves such that
sup llduvllL'"' < oo
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