论文标题
$ \ mathbb {r}^n $和$ \ mathbb {s}^n $上的等级气泡的结构
The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$
论文作者
论文摘要
1990年代的$ n $维欧几里得和球形空间中的多泡同性猜想断言，标准气泡在所有$ q-1 $ bubbles inclated offerscipt ofercript量中唯一最小化了总周边，任何$ q \ leq n+2 $。 Hutchings-Morgan-Ritoré-Ros在2000年确认了$ \ Mathbb {r}^3 $上的双重猜想，如今已完全解决了所有$ n \ geq 2 $。 $ \ mathbb {s}^2 $和$ \ mathbb {r}^2 $上的三重泡沫的猜想也已得到解决，但所有其他情况通常都开放。 We confirm the conjecture on $\mathbb{R}^n$ and on $\mathbb{S}^n$ for all $q \leq \min(5,n+1)$, namely: the double-bubble conjectures for $n \geq 2$, the triple-bubble conjectures for $n \geq 3$ and the quadruple-bubble conjectures for $n \geq 4 $。实际上，我们表明，对于所有$ q \ leq n+1 $，一个最小化的簇必定具有球形接口，在对$ \ mathbb {s}^n $的立体预测后，它的单元格是作为$ q $ offine offine offine offine offine offorme of $ \ mathbb y MathBB的Voronoi细胞获得的。 $ \ mathbb {r}^{n+1} $。此外，细胞（包括无限的细胞）必须连接并与对称性的常见超平面相交，从而解决了赫普斯的猜想。我们还显示了所有$ q \ leq n+1 $ $，表明所有对单元对之间具有非空接口的最小化器必然是标准气泡。证据使考虑$ \ mathbb {r}^n $和$ \ mathbb {s}^n $在串联和möbius几何和保形杀戮场中的关键使用至关重要。它并不依赖于为等级轮廓建立PDI，因为在当前的设置中似乎是遥不可及的。
The multi-bubble isoperimetric conjecture in $n$-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$. The double-bubble conjecture on $\mathbb{R}^3$ was confirmed in 2000 by Hutchings-Morgan-Ritoré-Ros, and is nowadays fully resolved for all $n \geq 2$. The double-bubble conjecture on $\mathbb{S}^2$ and triple-bubble conjecture on $\mathbb{R}^2$ have also been resolved, but all other cases are in general open. We confirm the conjecture on $\mathbb{R}^n$ and on $\mathbb{S}^n$ for all $q \leq \min(5,n+1)$, namely: the double-bubble conjectures for $n \geq 2$, the triple-bubble conjectures for $n \geq 3$ and the quadruple-bubble conjectures for $n \geq 4$. In fact, we show that for all $q \leq n+1$, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to $\mathbb{S}^n$, its cells are obtained as the Voronoi cells of $q$ affine-functions, or equivalently, as the intersection with $\mathbb{S}^n$ of convex polyhedra in $\mathbb{R}^{n+1}$. Moreover, the cells (including the unbounded one) are necessarily connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes. We also show for all $q \leq n+1$ that a minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble. The proof makes crucial use of considering $\mathbb{R}^n$ and $\mathbb{S}^n$ in tandem and of Möbius geometry and conformal Killing fields; it does not rely on establishing a PDI for the isoperimetric profile as in the Gaussian setting, which seems out of reach in the present one.
