Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces

Abstract

We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost (Two-dimensional geometric variational problems. Pure and applied mathematics. Wiley, New York, 1991) and Parker (J Differ Geom 44(3):595–633, 1996) respectively established these results by exploiting now classical arguments for harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. In the absence of a PDE, we take advantage of the local convexity properties of the target space. Included in this paper are an $$\epsilon $$ -regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image.