Improved Convergence Theorems for Bubble Clusters. II. The Three-Dimensional Case

Given a sequence {{ℰk}k of almost-minimizing clusters in ℝ3 that converges in L1 to a limit cluster ℰ, we prove the existence of C1,α-diffeomorphisms fk between ∂ℰ and ∂ℰk that converge in C1 to the identity. Each of these boundaries is divided into C1,α-surfaces of regular points, C1,α-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 ℝ3.