Bubble tree convergence for harmonic maps

Abstract

Let Σ be a compact Riemann surface. Any sequence fn : Σ → M of harmonic maps with bounded energy has a “bubble tree limit” consisting of a harmonic map f0 : Σ → M and a tree of bubbles fk : S2 → 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 fn converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps fn when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and (fn) is a Palais-Smale sequence for the harmonic map energy. © 1996 J. differential geometry.