The Gaussian Double-Bubble and Multi-Bubble Conjectures

• 成果类型: Article
• 署名作者: Milman, Emanuel; Neeman, Joe
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.195.1.2
• 发表日期: 2022
• 页码: 89-206
• 关键词: isoperimetric inequality; soap-bubble; minimizing perimeter; PROOF; REGULARITY; BOUNDARY; volume; THEOREMS; SPACE; sets

摘要

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussianweighted perimeter way to decompose Rn into q cells of prescribed (positive) Gaussian measure when 2 < q < n+ 1, is to use a simplicial cluster, obtained from the Voronoi cells of q equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to nullsets). In particular, the case q = 3 confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose Rn (n >= 2) into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an (n-2)-dimensional plane at 120 degrees angles (forming a tripod or Y shape in the plane). The case q = 2 recovers the classical Gaussian isoperimetric inequality. To establish the Multi-Bubble conjecture, we show that in the above range of q, stable regular clusters must have flat interfaces, therefore consisting of convex polyhedral cells (with at most q-1 facets). In the doublebubble case q = 3, it is possible to avoid establishing flatness of the interfaces by invoking a certain dichotomy on the structure of stable clusters, yielding a simplified argument.