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Paper   IPM / M / 16300
School of Mathematics
  Title:   Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces
  Author(s):  Sajjad Lakzian (Joint with Ch. Breiner)
  Status:   Published
  Journal: Calc. Var.
  Year:  2020
  Pages:   DOI: 10.1007/a00526-020-01801-w
  Supported by:  IPM
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 and Parker 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 ϵ-regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps for harmonic maps into metric spaces with upper curvature bounds. We also prove an isoperimetric inequality for conformal harmonic maps with small image.

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