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].

中文翻译：

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周长最小化平面和 2 球体中的三重气泡

摘要 我们使用连续和离散统一来证明 ℝ2 和 𝕊2 中的标准三重气泡是包围相同三重区域的所有簇（定义 2.3）中周长的最小值。统一定义了一个统一的衡量标准，允许所有配置，无论区域如何，都可以一起竞争。连续统一证明，如果统一的最小化器比预期的要好，那么它必须至少有一个内部气泡组件。离散统一证明内部气泡只能存在一个，而且它必须是连通的。只剩下“雏菊”配置：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。球体上的结果是新的；飞机上的结果是由于 Wichiramala [11]。球体中的双气泡由 Masters [6] 完成。