Improved convergence theorems for bubble clusters I. The planar case

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this construction to isoperimetric problems for planar bubble clusters. In this setting, we develop an improved convergence theorem, showing that a sequence of almost-minimizing planar clusters converging in L¹ to a limit cluster must actually converge in a strong C^1,α-sense. We also discuss applications of this improved convergence result to the classification of isoperimetric clusters and the qualitative description of perturbed isoperimetric clusters. In part two, we give analogous results for three-dimensional clusters; further applications are discussed in some companion papers.