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

TL;DR

This paper extends bubble tree convergence results for harmonic maps into compact locally CAT(1) spaces, demonstrating energy quantization and no-neck properties using geometric methods without PDE reliance.

Contribution

It provides a geometric reinterpretation of bubble tree convergence, energy quantization, and no-neck properties for harmonic maps into CAT(1) spaces, including new regularity and gap theorems.

Findings

Established energy quantization for harmonic maps into CAT(1) spaces.

Proved no-neck property for sequences of harmonic maps.

Developed regularity and gap theorems for harmonic maps into metric 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 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 $\epsilon$-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.