Improved Convergence Theorems for Bubble Clusters. Ii. the Three-dimensional Case

Given a sequence {Ek}k of almost-minimizing clusters in R which converges in L to a limit cluster E we prove the existence of C-diffeomorphisms fk between ∂E and ∂Ek which converge in C to the identity. Each of these boundaries is divided into C-surfaces of regular points, C-curves of points of type Y (where the boundary blows-up to three half-spaces meeting along a line at 120 degree) and isolated points of type T (where the boundary blows up to the two-dimensional cone over a one-dimensional regular tetrahedron). The diffeomorphisms fk are compatible with this decomposition, in the sense that they bring regular points into regular points and singular points of a kind into singular points of the same kind. They are almost-normal, meaning that at fixed distance from the set of singular points each fk is a normal deformation of ∂E, and at fixed distance from the points of type T , fk is a normal deformation of the set of points of type Y . Finally, the tangential displacements are quantitatively controlled by the normal displacements. This improved convergence theorem is then used in the study of isoperimetric clusters in R.