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 **[2]**).

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 **[1]** 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**

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

**[2]** McDuff and Salamon, *J-Holomorphic curves and Symplectic Topology*