Bubble Tree Convergence for Harmonic Maps

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 /o : Σ -> M and a tree of bubbles fk : 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 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 {/n} is a Palais-Smale sequence for the harmonic map energy. Consider a sequence of harmonic maps fn : Σ —> M from a compact Riemann surface (Σ, h) to a compact Riemannian manifold (M,g) with bounded energy (0.1) E(fn) = \JΣ\dfn\ 2 < Eo. Such a sequence has a well-known "Sacks-Uhlenbeck" limit consisting of a harmonic map /o : Σ —> M and some "bubbles" — harmonic maps S -> M obtained by a renormalization process. In fact, by following the procedure introduced in [12], one can modify the Sacks-Uhlenbeck renormalization and iterate, obtaining bubbles on bubbles. The set of all bubble maps then forms a "bubble tree" ([12]). One would like to know in precisely what sense the sequence {fn} converges to this bubble tree. The major issue is the appearance of "necks" joining one bubble to the next. Received August 29, 1994, and, in revised form, August 14, 1996.