Growth of Energy in Minimal Surface Bubbles

This article is concerned with developing tools for investigating harmonic maps f : Σ→M from a closed Riemann surface Σ into a compact manifold M of dimension at least three, using a perturbative approach based upon the α-energy of Sacks and Uhlenbeck. We present a replacement procedure for α-harmonic maps which is similar to one that has been used for harmonic maps, and show how it can be used to investigate the structure of critical points for the α-energy, when α > 1 is sufficiently close to one. We give an estimate on the rate of growth of the energy density in the bubbles of such critical points as α→ 1, when the bubbles are at a distance at least L0 > 0 from the base.