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

摘要

Given a sequence {E-k}(k) of almost-minimizing clusters in R-3 that converges in L-1 to a limit cluster E, we prove the existence of C-1,C- alpha-diffeomorphisms f(k) between partial derivative E and partial derivative E-k that converge in C-1 to the identity. Each of these boundaries is divided into C-1,C- alpha-surfaces of regular points, C-1,C- alpha-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 onedimensional regular tetrahedron). The diffeomorphisms f(k) 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 f(k) is a normal deformation of partial derivative 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-3.