Energy Growth in Minimal Surface Bubbles

This article is concerned with conformal harmonic maps f : Σ → M from a closed connected surface Σ into a compact Riemannian manifoldM of dimension at least four, studied via a perturbative approach based upon the α-energy of Sacks and Uhlenbeck. It gives an estimate on the rate of growth of energy density in the bubbles of minimax α-energy critical points as α→ 1, when the bubbles are at a distance at least L0 > 0 from the base. It also describes additional techniques hopefully useful for the development of a partial Morse theory for closed parametrized minimal surfaces (or harmonic surfaces) in compact Riemannian manifolds.