It is shown that m disjoint sets with fixed Gaussian volumes that partition \(\mathbb {R}^{n}\) with minimum Gaussian surface area must be \((m-1)\)-dimensional. This follows from a second variation argument using infinitesimal translations. The special case \(m=3\) proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when \(m=3\), the three minimal sets are adjacent 120 degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove the \(m=4\) Triple Bubble Conjecture for the Gaussian measure. Our methods combine the Colding–Minicozzi theory of Gaussian minimal surfaces with some arguments used in the Hutchings–Morgan–Ritoré-Ros proof of the Euclidean Double Bubble Conjecture.

结果表明，具有固定高斯体积的m个不相交集将\（\ mathbb {R} ^ {n} \）划分为最小高斯表面积，必须是\（（m-1）\）-维。这源自使用无穷小转换的第二个变元。特殊情况\（m = 3 \）证明了高斯测度的双重气泡问题，并带有额外的技术假设。也就是说，当\（m = 3 \）时，这三个最小集合是相邻的120度扇区。技术上的假设是，最小化集合的三重结点具有多项式体积增长。再次假设技术假设，我们证明\（m = 4 \）高斯测度的三重气泡猜想。我们的方法将高斯最小曲面的Colding-Minicozzi理论与欧几里德双气泡猜想的Hutchings-Morgan-Ritoré-Ros证明中使用的一些论点结合在一起。