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In the paper “Bubble Tree Convergence for Harmonic Maps” by Thomas H. Parker, after the picking the energy concentration points, he proceeded by expanding the map around each energy concentration point and then define the renormalized map by doing a translation of the centre of mass of the energy measure $e(f_n)$ in a small ball and then a dilation by a sequence $\lambda_n\to 0$. Similar argument is also done in Appendix B6 in “Width and Finite Extinction Time” by Colding and Minicozzi.

When I inspect the proof in these two paper, I see that the argument still goes through if we just do the dilation centred at each energy concentration point instead of considering centre of mass of the energy measure $e(f_n)$. Am I missing something or is there any specific reason we have to consider off-centre balls instead of the balls centred at the energy concentration points?

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    $\begingroup$ The issue is that the scale of the bubble might be much smaller than the distance of the center of mass to the "energy concentration point" . Think of the bubble as living in $B_{\epsilon^2}(\epsilon,0)$. Then, if you dilate around $(0,0)$, you can never actually see the bubble, you will either see a smoothly converging sequence, or a sequence that again has an energy concentration point. The exact choice of center of mass is not important, but you do need some choice that allows you to optimize the center of the rescaling. $\endgroup$
    – Otis Chodosh
    Commented Apr 3, 2022 at 20:38
  • $\begingroup$ @OtisChodosh Correct me if I am wrong. To my understanding, around an energy concentration point $x$, there are further “bubbles” $B_{r_n}(x_n)$ with $r_n \to 0$ and $x_n \to x$ and $f_n|_{B_{r_n}(x_n)}$ have a definite amount of energy (at least $\epsilon_0$). To keep these new bubbles at the south pole, we can dilate the maps at the smallest ball containing these bubbles. However, we may push every bubbles to the south pole because of the bad choice of centre. Therefore, we need choose an alternative centre is to control the radius of the balls containing these new bubbles. $\endgroup$
    – Loafy Loafer
    Commented Apr 4, 2022 at 8:22
  • $\begingroup$ I am not sure I 100% understood, but I think yes this is the idea. If you don't recenter you may never be able to actually see the bubble map. It's a good exercise to work this out with the identity map $S^2\to S^2$. Then perform a conformal dilation around some point $x_n$. Try to see how you would see the bubble forming, e.g. what happens if you just dilate around the south pole, etc. $\endgroup$
    – Otis Chodosh
    Commented Apr 4, 2022 at 14:44

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