The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866910914431680512 |
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| author | Milman, Emanuel Neeman, Joe |
| author_facet | Milman, Emanuel Neeman, Joe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_09102 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$ Milman, Emanuel Neeman, Joe Differential Geometry Functional Analysis Metric Geometry 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. |
| title | The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$ |
| topic | Differential Geometry Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2205.09102 |