Bubbling of Almost-harmonic Maps between 2-spheres at Points of Zero Energy Density

Abstract

We show that bubbling of almost-harmonic maps between 2-spheres has very different behaviour depending on whether or not bubbles develop at points in the domain at which the energy density of the body map is zero. We also see that this translates into different behaviour for the harmonic map flow. In [11] we obtained results, assuming nonzero bubble point density for certain bubbles, forcing the harmonic map flow to converge uniformly and exponentially to its limit. This involved proving a type of nondegeneracy for the harmonic map energy (a ‘quantization’ estimate’) and an estimate on certain bubble scales (a ‘repulsion’ estimate). Here we show that without the nonzero bubble point density hypothesis, both the quantization and repulsion estimates fail, and we construct a flow in which the convergence is no longer exponentially fast.