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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

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  • $\begingroup$ @ChrisGerig yeah I heard this slogan, but what if I don't have any marked point to begin with? my impression with paper [1] is that there is no assumptions on marked point. Hence the confusion. $\endgroup$ – Riccardo Feb 24 '19 at 21:53
  • $\begingroup$ Can you elaborate a little bit on why the ghost bubbles are bounded above by the marked points? $\endgroup$ – Riccardo Feb 24 '19 at 21:54
  • $\begingroup$ @ChrisGerig I guess I'm misunderstanding the paper [1] then :( $\endgroup$ – Riccardo Feb 24 '19 at 22:45
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Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.

Look at Lemma 4.2 in Parker-Wolfson and how it is used later on (pg. 91-92): Their ghost bubbles sit in a sequence which must terminate at a bubble/component having energy concentration, so the bubble tree is still finite. Note that if each step of the bubbling process produced infinitely many ghost bubbles then the curve (bubble tree) would be noncompact.

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).

This is also discussed in Parker's AMS notice "What is... a Bubble Tree?".

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  • $\begingroup$ 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. $\endgroup$ – Riccardo Feb 25 '19 at 17:58
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    $\begingroup$ The iteration occurs because we are trying to find out where the energy is really going/concentrating. Consider the base case: If no energy is concentrating at a point along the sphere, then no ghost bubble is going to form there, the renormalization process does nothing. It’s only when you had energy concentrating at some point that you’ll get a bubble tree based at that point, so the concentrated energy is now somewhere in that tree and we repeat the process. $\endgroup$ – Chris Gerig Feb 25 '19 at 18:39

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