The Gaussian Double-Bubble and Multi-Bubble Conjectures

Abstract

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.

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      mrreviewer = {A. Badrikian},
      doi = {10.7146/math.scand.a-12035},
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      author = {Ehrhard, Antoine},
      title = {\'{E}léments extrémaux pour les inégalités de {B}runn-{M}inkowski gaussiennes},
      journal = {Ann. Inst. H. Poincaré Probab. Statist.},
      fjournal = {Annales de l'Institut Henri Poincaré. Probabilités et Statistique},
      volume = {22},
      year = {1986},
      number = {2},
      pages = {149--168},
      issn = {0246-0203},
      mrclass = {60E15 (60G17)},
      mrnumber = {0850753},
      mrreviewer = {R. M. Dudley},
      url = {http://www.numdam.org/item?id=AIHPB_1986__22_2_149_0},
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      volume = {153},
      publisher = {Springer-Verlag New York Inc., New York},
      year = {1969},
      pages = {xiv+676},
      mrclass = {28.80 (26.00)},
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      title = {Soap bubble clusters in {$\mathbb{R}^2$} and {$\mathbb{R}^3$}},
      note = {{S}enior {H}onors {T}hesis, {W}illiams {C}ollege, Williamstown, MA},
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      author = {Foisy, Joel and Alfaro, Manuel and Brock, Jeffrey and Hodges, Nickelous and Zimba, Jason},
      title = {The standard double soap bubble in {${\bf R}^2$} uniquely minimizes perimeter},
      journal = {Pacific J. Math.},
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      year = {1993},
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      mrclass = {53A10 (49Q05)},
      mrnumber = {1211384},
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      doi = {10.2140/pjm.1993.159.47},
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      doi = {10.1090/S1079-6762-95-03001-0},
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      mrclass = {60E15 (53A10 58E30 60G15)},
      mrnumber = {4267646},
      doi = {10.1007/s12220-020-00531-x},
      url = {https://doi.org/10.1007/s12220-020-00531-x},
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      title = {Solution of the propeller conjecture in {$\Bbb{R}^3$}},
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      pages = {195--200},
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      year = {2003},
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      doi = {10.1007/978-3-642-55627-2_12},
      url = {https://doi.org/10.1007/978-3-642-55627-2_12},
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      mrreviewer = {Werner Linde},
      doi = {10.4064/sm-118-2-169-174},
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      volume = {135},
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      year = {2012},
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      isbn = {978-1-107-02103-7},
      mrclass = {49-01 (26B20 28-02 49-02 49Q05 49Q20)},
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      doi = {10.1017/CBO9781139108133},
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      journal = {Geom. Dedicata},
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      }
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      mrclass = {53A10 (49Q10)},
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      doi = {10.4310/AJM.2001.v5.n1.a3},
      url = {https://doi.org/10.4310/AJM.2001.v5.n1.a3},
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      title = {Soap bubbles in {${\bf R}^2$} and in surfaces},
      journal = {Pacific J. Math.},
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      mrclass = {58E12 (49Q05 53A10)},
      mrnumber = {1300837},
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      doi = {10.2140/pjm.1994.165.347},
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      journal = {Trans. Amer. Math. Soc.},
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      title = {Manifolds with density},
      journal = {Notices Amer. Math. Soc.},
      fjournal = {Notices of the American Mathematical Society},
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      author = {Morgan, Frank},
      title = {Geometric Measure Theory},
      edition = {Fifth},
      titlenote = {A beginner's guide, illustrated by James F. Bredt},
      publisher = {Elsevier/Academic Press, Amsterdam},
      year = {2016},
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Authors

Emanuel Milman

Technion - Israel, Institute of Technology, Haifa, Israel

Joe Neeman

University of Texas at Austin, Austin, TX, USA