Proof of the Double Bubble Conjecture
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Double Bubbles on the Real Line with Log-Convex Density
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).
math.DGmath.MG
Double Bubbles with High Constant Mean Curvatures in Riemannian Manifolds
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.
math.APmath.DG
平稳变分的双泡与三泡问题：凸情形
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。
math.APmath.DG
具有孤立气泡的调和映射流解的Topping猜想
Topping's conjecture for solutions of harmonic map flow with isolated bubbles
本文通过建立加权Lojasiewicz-Simon不等式，证明了调和映射流在仅形成孤立气泡的有限时间奇点情形下的体映射连续性的Topping猜想。
math.APmath.DG
Double bubbles for immiscible fluids in $\mathbb{R}^n$
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.
math.DGmath.OC
Double bubbles in the 3-torus
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.
math.DGmath.GT
更多相关论文收起更多论文
Grushin平面中的对称双气泡
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。
math.DGmath.MGmath.OC
$\mathbb{R}^n$ 与 $\mathbb{S}^n$ 上等周气泡的结构
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。
math.DGmath.FAmath.MG
Double Bubbles Minimize
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.
math.DGmath.OC
Double bubbles in $S^3$ and $H^3$
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.
math.DG
The Gaussian Double-Bubble and Multi-Bubble Conjectures
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 (…
math.DGmath.FAmath.PR
平台气泡与 $\mathbb{S}^n$ 上的五重气泡定理
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。
math.DGmath.FAmath.MG
The Gaussian Double-Bubble Conjecture
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).
math.DGmath.FAmath.PR
无穷双气泡定理
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。
math.APmath.DGmath.MG
非单连通双核心猜想的反例
Counterexamples to the non-simply connected Double Soul Conjecture
研究非单连通双核心猜想，通过构造了反例，无穷多个3维反例，以及维度无界增长的平坦反例，我。
math.DG
Stable Gaussian Minimal Bubbles
It is shown that $3$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with nearly minimum total Gaussian surface area must be close to adjacent $120$ degree sectors, when $n\geq2$. These same results hold for any number $m\leq n+1$ of sets partitioning $\mathbb{R}^{n}$, conditional on the solution of a finite-dimensional optimization problem (similar to the endpoint case of the Plurality is Stablest Problem, or the Propeller Conjecture of Khot and Naor). When $m>3$, the minimal Gaussian surface area is achieved by the cones over a regular simplex. We therefore strength…
math.DGmath.FAmath.PR
