Geometric Analysis and Topology Seminar
Isoperimetric Multi-Bubble Problems - Old and New
Speaker: Emanuel Milman, Technion
Location: Online
Videoconference link: https://nyu.zoom.us/j/94747124588
Date: Wednesday, April 27, 2022, 11 a.m.
Synopsis:
The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$, known to the ancient Greeks in dimensions two and three, states that among all sets (``bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the sphere $\mathbb{S}^n$ and on Gauss space $\mathbb{G}^n$. Furthermore, one may consider the ``multi-bubble" partitioning problem, where one partitions the space into $q \geq 2$ (possibly disconnected) bubbles, so that their total common surface-area is minimal. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $q=2$; the case $q=3$ is called the double-bubble problem, and so on.
In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the Double-Bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) -- the optimal partition into two bubbles of prescribed finite volumes (and an exterior unbounded third bubble) which minimizes the total surface-area is given by three spherical caps, meeting at 120-degree angles. A more general conjecture of J.~Sullivan from the 1990's asserts that when $q \leq n+2$, the optimal Multi-Bubble partition of $\mathbb{R}^n$ (as well as $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $q$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards).
In 2018, together with Joe Neeman, we resolved the analogous Multi-Bubble conjecture on the optimal partition of Gauss space $\mathbb{G}^n$ into $q \leq n+1$ bubbles -- the unique optimal partition is given by the Voronoi cells of (appropriately translated) $q$ equidistant points. In this talk, we will describe our approach in that work, as well as recent progress on the Multi-Bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing partitions are always spherical when $q \leq n+1$, and we resolve the latter conjectures when in addition $q \leq 6$ (e.g. the triple-bubble conjecture in $\mathbb{R}^3$ and $\mathbb{S}^3$, and the quadruple-bubble conjecture in $\mathbb{R}^4$ and $\mathbb{S}^4$).
Based on joint work (in progress) with Joe Neeman.