# Intuition about bubbling off a ghost bubble

I'm trying to improve my intuition about the bubbling phenomenon for $$J$$-holomorphic curves $$\Sigma \to (M,\omega)$$, where $$\Sigma$$ is a compact Riemann surface with possibly boundary. I assume that the complex structure $$j_{\Sigma}$$ and Riemann metric $$\text{dvol}_{\Sigma}$$ is fixed once and for all.

My understanding of the bubbling phenomenon is closely related to the construction by Uhlenbeck and Sachs where given a sequence $$\{ f_n\}_n \colon \Sigma \to M$$ of $$J$$-holomorphic curves with uniformity bounded energy but not uniformly bounded $$\|df_n\|_{L^{\infty}(\Sigma)}$$ we can construct a sequence of $$J$$-hol maps $$g_n \colon S^2 \to M$$ which converges uniformly to a map $$g\colon S^2\to M$$ (I'm referring to chapter 4.2 in ).

If I understood things correctly in this procedure there must be a potential loss of energy, because it would prevent the creation of bubbles on bubbles, which instead happens. Is that correct?

In fact in  it's claimed that a more careful renormalization procedure is necessary and in fact they construct a new sequence of $$J$$-holomorphic spheres $$\{g_n'\}_n$$ that uniformly converges on every compact $$K\subset S^2\setminus \{y_1,\dots y_l\}$$ to a $$J$$-hol sphere $$g$$ and then if you iterate the procedure on these points $$y_1,\dots y_l$$ you might have to attach bubbles on there as well. So far so good, only problem is that they claim (page 85) that

It is possible for $$g$$ to have zero energy, i.e. $$g$$ is a constant map.

How should I think of this ghost bubble? What can prevent me from having infinitely many ghost bubbles in my tree? (the usual argument with energy won't work since they don't carry any energy). They claim that every step of their procedure reduces the energy by at least $$\hslash$$ (the lower bounds on energy for spheres or disks with Lagrangian boundary conditions), but I don't see why in case of ghost bubbles.

Can someone explain that to me?

Reference

 Parker and Wolfson, Pseudo-Holomorphic Maps and Bubble Trees, The Journal of Geometric Analysis, vol. 3, Number 1, 1993

 McDuff and Salamon, J-Holomorphic curves and Symplectic Topology

• @ChrisGerig yeah I heard this slogan, but what if I don't have any marked point to begin with? my impression with paper  is that there is no assumptions on marked point. Hence the confusion. – Riccardo Feb 24 '19 at 21:53
• Can you elaborate a little bit on why the ghost bubbles are bounded above by the marked points? – Riccardo Feb 24 '19 at 21:54
• @ChrisGerig I guess I'm misunderstanding the paper  then :( – Riccardo Feb 24 '19 at 22:45

Restating, if a stable ghost bubble in a bubble tree has less than 3 marked points then: It is either at the end of a tree branch with 2 marked points (and 1 node), or it is in the middle of a chain with 1 or 2 marked points (and 2 nodes), or it is a connector'' for at least 3 other components (i.e. it has at least 3 nodes).
• dear Chris, thanks a lot for the clarifying answer. I must apologise in advance but I still don't see why I can't just keep on attaching ghost bubbles to ghost bubbles. I think the problem in my understanding is that, in the presence of Ghost bubbles, I don't clearly see why every iteration of this procedure costs $\hslash$ energy. Because if the procedure has to end, by lemma 4.2 I agree that eventually there must be a non-constant bubble that "ends" the branch of my tree. – Riccardo Feb 25 '19 at 17:58