Remarks on the existence of branch bubbles on the blowup analysis of equation \(-\Delta u= e^{2u}\) in dimension two (Q1293490)

It is well known that for a sequence of metrics in a fixed conformal class with constant scalar curvature on a manifold with dimension 3 or higher, the blowingup set must be finite and simple (each bubbling point carry exactly one sphere). The corresponding statement in dimension 2 is widely expected to hold. The purpose of this paper is to construct an example of a sequence of metrics in a unit disk with constant curvature 1 and uniformly bounded area which develops branch bubbles at the center of the disk, thereby providing a counter example to the analogous statements in dimension two.