The Gaussian double-bubble and multi-bubble conjectures. (English) Zbl 1484.49072

Milman, Emanuel; Neeman, Joe

Ann. Math. (2) 195, No. 1, 89-206 (2022).

Summary: We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose \(\mathbb{R}^n\) into \(q\) cells of prescribed (positive) Gaussian measure when \(2\leq q\leq n+1\), 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 null-sets). In particular, the case \(q=3\) confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose \(\mathbb{R}^n\) \((n\geq 2)\) 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 \(120^{\circ}\) 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 double-bubble 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.

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

isoperimetric problem; isoperimetric profile; Gaussian measure; double-bubble; multi-bubble; stable clusters

References:

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