Narrow Results

We generalize the descriptions of vortex moduli spaces in [4] to more than one section with adiabatic constant $s$. The moduli space is topologically independent of $s$ but is not compact with respect to $C^{\infty }$ topology. Following [17], we construct a Gromov limit for vortices of fixed energy, and attempt to compactify the moduli space via bubble trees with possibly conical bubbles (or raindrops).

We give a description of singularity formation in terms of energy quanta for 2-dimensional radially symmetric equivariant harmonic map heat flows. Adapting Struwe's energy method we first establish a finite bubble tree result with a discrete multiple of energy quanta disappearing in the singularity. We then use intersection-comparison arguments to show that the bubble tree consists of a single bubble only and that there is a well defined scale R_BHK(t) ↓ 0 in which the solution converges to the standard harmonic map.