Abstract
Given a sequence $\{\mathcal{E}_{k}\}_{k}$ of almost-minimizing clusters in $\mathbb{R}^3$ which converges in $L^{1}$ to a limit cluster $\mathcal{E}$ we prove the existence of $C^{1,\alpha}$-diffeomorphisms $f_k$ between $\partial\mathcal{E}$ and $\partial\mathcal{E}_k$ which converge in $C^1$ to the identity. Each of these boundaries is divided into $C^{1,\alpha}$-surfaces of regular points, $C^{1,\alpha}$-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 one-dimensional 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 $\partial\mathcal{E}$, and at fixed distance from the points of type $T$, $f_k$ 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 $\mathbb{R}^3$.
Highlights
Given a sequence {exists xk ∈ ΣT (Ek)}k of almost-minimizing clusters in R3 which converges in L1 to a limit cluster E we prove the existence of C1,α-diffeomorphisms fk between ∂E and ∂Ek which converge in C1 to the identity
We introduce an equivalence relation ≈ on the family of clusters in R3 that are (Λ, r0)-minimizing cluster for some choice of Λ ≥ 0 and r0 > 0, by setting E ≈ F if and only if there exists a C1,α-diffeomorphism between ∂E and ∂F
Step one: We show that to each γ ∈ ΓY (E) and k ≥ k0 one can associate γk ∈ ΓY (Ek) in such a way that
Summary
The goal of this section is showing the convergence of singular sets and tangent cones for clusters in R3. Since X′ = Tx∂E satisfies θX′ (0) = θ∂E (x) = θ∂Ek (x), by Theorem A-(ii) and (3.12) we will deduce the existence of minimal cones Xk such that if k ≥ kx, hd0,1(Xk, X′) ≤ η , hdx,r(∂Ek, x r. By (3.16), the inclusion spt(Γrj ) ⊂ Sj ∩ f −1(r) and the definition of A3, there exists a C1-curve with boundary γ such that, if Tγ denotes the one-dimensional multiplicity-one integral current associated with (one of the two orientations of) γ, . We may assume, arguing by contradiction, that x ∈ ΣY (E) If this is the case, there exists rx > 0 and an injective map σ : {1, 2, 3} → {0, ..., N } such that |E(h) ∩ Bx,rx| = 0 if h = σ(i), i = 1, 2, 3. We have reached a contradiction, and completed the proof of the theorem
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