Joel Hass, Roger Schlafly
The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 degrees.
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We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem. As part of the proof we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.
We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.
The double and triple bubble problem for stationary varifolds: the convex case
该研究针对凸气泡情形，刻画了n维欧氏空间中双泡问题与三维欧氏空间中三泡问题的临界点，推进了几何测度论相关问题研究。
We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \Bbb R^3.
The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we assume to be strictly log-convex. For $N=1$ we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions, we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).
Symmetric double bubbles in the Grushin plane
本文研究Grushin平面中两等体积气泡的双气泡问题，通过对称化与一阶变分证明极小元存在并刻画其性质，发现垂直与水平界面极小元周长差异，并猜想后者为无假设下的最优解。
We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.
We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of $S^3$; 2) in $H^3$, the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem in $S^3$ and $H^3$ to some computer checking. The computer analysis has been designed and fully implemented for both spaces.
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). In particular, the case $q=3$ confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ ($n \geq 2$) into three cells of prescribed (…
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ angles (forming a tripod or "Y" shape in the plane). Moreover, we prove that tripod-clusters are the unique isoperimetric minimizers (up to null-sets).
Multi-Bubble Isoperimetric Problems
本文综述多泡等周极小化子刻画与肥皂泡分块稳定性的最新进展，提出相关开放问题，为该领域研究提供参考。
Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome.
We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in ${\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact mani…
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 …
An Infinite Double Bubble Theorem
本文证明了双气泡定理的一个变体，其中两个腔室具有无穷体积，证明了加权透镜簇是局部最小化的，并在特定对称性和增长假设下确立了其唯一性，同时得到了局部最小化 (N,2)-cluster 的闭包定理。
The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$
本文证明了欧氏空间和球面上对所有$q \leq \min(5,n+1)$的多气泡等周猜想，并刻画了极小化簇的球面界面和Voronoi结构，同时解决了Heppes猜想。
This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which $\mathbb{R}^N$ gets partitioned. We then consider the problem of characterizing the sets $Ω$ that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of $Ω$ in the two half-spaces are given. It is sho…
Plateau Bubbles and the Quintuple Bubble Theorem on $\mathbb{S}^n$
本文研究球面与欧氏空间中的多重气泡等周问题，通过发展Jacobi算子谱理论和新的气泡形变方法，证明了S^n（n≥5）上的五重气泡猜想，并在Plateau定律高维类比假设下解决q≤n+1的情形，同时证明最小化构型Jacobi指标恰为q-1。
We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we present a compactness result for branched, immersed, haunted, stratified surface with bounded area and Willmore energy. This allows us to prove the existence of area constrained minimizers for generalized Willmore functionals in the class of haunted, branched, immersed bubble trees by direct minimization. Here a haunted, stratified surfaces are introduced, …
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