Bubble tree convergence for harmonic maps

Abstract

Let be a compact Riemann surface. Any sequence f n : -> M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : -> M and a tree of bubbles f k : S 2 -> M. We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the f n converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps f n when the conformal structure of varies with n, and (ii) when the conformal structure is fixed and {/ n } is a Palais-Smale sequence for the harmonic map energy.