Perimeter-Minimizing Triple Bubbles in the Plane and the 2-Sphere

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2019-01-01 DOI:10.1515/agms-2019-0004

G. Lawlor

引用次数: 5

Abstract

Abstract We use continuous and discrete unification to prove that standard triple bubbles in ℝ2 and 𝕊2 are the minimizers of perimeter, among all clusters (Definition 2.3) enclosing the same triple of areas. Unification defines a unified measurement that allows all configurations, regardless of areas, to compete together. Continuous unification proves that if a unified minimizer were better than expected, it would have to have at least one interior bubble component. Discrete unification proves there can only be one interior bubble and that it must be connected. This leaves only the “daisy” configurations: one interior bubble surrounded by an even number of “petals.” A more careful analysis also eliminates these, leaving only the standard triple bubbles as minimizers. The result on the sphere is new; the result in the plane is due to Wichiramala [11]. The double bubble in the sphere was done by Masters [6].

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最小化平面和双球面中三个气泡的周长

摘要利用连续和离散统一证明了在包含相同区域的所有簇(定义2.3)中，标准三重泡(𝕊2)是周长的最小值。统一定义了一个统一的度量，允许所有的配置，不管区域，一起竞争。连续统一证明，如果一个统一的最小化器比预期的要好，它必须至少有一个内部气泡组件。离散统一证明了内部气泡只能有一个，而且它必须是相互连接的。这就只剩下了“雏菊”的结构:一个内部气泡被偶数个“花瓣”包围。更仔细的分析也消除了这些，只留下标准的三重气泡作为最小化。球面上的结果是新的;在飞机上的结果是由于威奇拉马拉[11]。球体上的双泡是马斯特斯做的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.