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.

我们确定从均匀的Riemann曲面到紧凑的局部CAT（1）空间中具有均匀能量范围的一系列谐波映射的气泡树收敛。特别是，我们展示了这种序列的能量量化和无颈特性。在平滑的环境中，Jost（二维几何变分问题。纯粹数学和应用数学。Wiley，纽约，1991年）和Parker（J Differ Geom 44（3）：595-633，1996年）分别利用现在的方法建立了这些结果。调和图的经典论证。我们的工作表明，这些结果可以用几何方法重新解释。在没有PDE的情况下，我们利用了目标空间的局部凸性。本文包括一个\（\ epsilon \）正则定理，能隙定理和谐波映射到具有上曲率边界的度量空间的可移动奇异性定理。我们还证明了具有小图像的共形谐波图的等距不等式。