{"paper":{"title":"Improved convergence theorems for bubble clusters. II. The three-dimensional case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OC"],"primary_cat":"math.AP","authors_text":"Francesco Maggi, Gian Paolo Leonardi","submitted_at":"2015-05-25T18:25:19Z","abstract_excerpt":"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-d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06709","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}