[Paper Review] The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$

Emanuel Milman, Joe Neeman|arXiv (Cornell University)|May 18, 2022

Point processes and geometric inequalities4 citations

TL;DR

This paper resolves the multi-bubble isoperimetric conjecture for $\mathbb{R}^n$ and $\mathbb{S}^n$ for all $q \leq \min(5, n+1)$, proving that minimizers have spherical interfaces and are equivalent to Voronoi cells of $q$ affine functions in $\mathbb{R}^{n+1}$, or convex polyhedral intersections with $\mathbb{S}^n$. The proof leverages Möbius geometry and conformal Killing fields, confirming that cells are connected, symmetric, and form standard bubbles when all interfaces are non-empty.

ABSTRACT

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.

Motivation & Objective

• To resolve the multi-bubble isoperimetric conjecture for $\mathbb{R}^n$ and $\mathbb{S}^n$ for $q \leq \min(5,n+1)$, where $q$ is the number of cells.
• To characterize the structure of isoperimetric clusters as Voronoi cells of affine functions or convex polyhedral intersections with $\mathbb{S}^n$.
• To prove that minimizing clusters have spherical interfaces and are connected, intersecting a common hyperplane of symmetry, resolving a conjecture of Heppes.
• To establish that minimizers with non-empty interfaces between all pairs of cells are standard bubbles.

Proposed method

• Utilizes Möbius geometry and conformal Killing fields to relate isoperimetric structures on $\mathbb{R}^n$ and $\mathbb{S}^n$ via stereographic projection.
• Analyzes the geometry of $q$-clusters using weighted Riemannian manifolds with smooth positive densities, focusing on $\mu$-weighted perimeter and volume.
• Applies the concept of Voronoi cells of $q$ equidistant points in $\mathbb{S}^n \subset \mathbb{R}^{n+1}$ to define standard bubbles.
• Employs the second fundamental form $\mathrm{I\!I}$ and weighted tangential divergence $\text{div}_{\Sigma,\mu}$ to compute curvature and mean curvature terms.
• Uses the Codazzi equation and weighted Ricci tensor $\text{Ric}_{\Sigma,\mu} = \text{Ric}_{\Sigma} + \nabla^2 W$ to derive geometric identities in the proof of Lemma B.1.
• Performs a technical computation in Riemannian normal coordinates to verify the divergence identity for $\text{div}_{\Sigma,\mu}(X^{\mathfrak{n}}\mathrm{I\!I}Y^{\mathbf{t}})$.

Experimental results

Research questions

• RQ1 Are isoperimetric $q$-clusters on $\mathbb{R}^n$ and $\mathbb{S}^n$ uniquely realized as standard bubbles for $q \leq \min(5,n+1)$?
• RQ2 Do minimizing clusters on $\mathbb{R}^n$ and $\mathbb{S}^n$ have spherical interfaces and arise as intersections of $\mathbb{S}^n$ with convex polyhedra in $\mathbb{R}^{n+1}$?
• RQ3 Is the cell structure of minimizers connected and symmetric, intersecting a common hyperplane of symmetry?
• RQ4 Under what conditions is a minimizer with all pairwise interfaces non-empty necessarily a standard bubble?

Key findings

• The multi-bubble isoperimetric conjecture is confirmed for $\mathbb{R}^n$ and $\mathbb{S}^n$ when $q \leq \min(5,n+1)$, covering double-bubble for $n \geq 2$, triple-bubble for $n \geq 3$, and quadruple-bubble for $n \geq 4$.
• Minimizing clusters on $\mathbb{R}^n$ and $\mathbb{S}^n$ for $q \leq n+1$ have spherical interfaces and are equivalent to Voronoi cells of $q$ affine functions in $\mathbb{R}^{n+1}$.
• After stereographic projection, the cells of a minimizer on $\mathbb{R}^n$ are the intersections of $\mathbb{S}^n$ with convex polyhedra in $\mathbb{R}^{n+1}$.
• All cells, including the unbounded one, are connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes.
• A minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble for $q \leq n+1$.
• The proof relies on Möbius geometry and conformal Killing fields, avoiding the use of PDI methods from the Gaussian setting, which are inapplicable here.