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

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
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### References:

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