arXiv research

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 sometime…

The paper proves an infinite double bubble theorem in higher dimensions.

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…

Characterizes critical points in convex double and triple bubbles.

Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.

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.

An elementary proof found for the double bubble problem in a specific norm.

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.

Hexagonal norm double bubble problem solved with minimal configurations.

Solves the quintuple bubble problem on spheres and Euclidean spaces.

We address the double bubble problem for the anisotropic Grushin perimeter $P_α$, $α\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the case of two equal volumes. We assume that the contact interface between the bubbles lays on either the vertical or the horizontal axis. Since no regularity theory is available i…

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 ang…

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…

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,…

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 $120^{\circ}$ …

Confirms isoperimetric conjectures on R^n and S^n for q ≤ min(5, n+1).

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 balanc…

Bubbles are essential in certain economic models with high growth and low interest rates.

Rational bubbles form in nonstationary models of real assets.

By using Moser's iteration technique, we show some removable singularity theorem of the tension field for biharmonic maps into manifolds of non-positive curvature, and the bubbling theorem of biharmonic maps and also harmonic maps.

The supplement proves the existence and properties of a dynamical system related to asset price bubbles.

Continuous time analysis of bubble formation in harmonic maps.

Three theorems about arbitrage bubbles in financial equations.

Develops Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…

The study proves manifold properties related to positive scalar curvature.

Can you fill R^n with a froth of "soap bubbles" that meet at most n at a time? Not if they have bounded diameter, as follows from Lebesgue's Covering Theorem. We provide some related results and conjectures.

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper…

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 volum…

Gluing theorem for collapsing warped-QAC Calabi-Yau manifolds verified.

The paper proves rigidity results for Serrin-type problems in manifolds.

Compactness theorem for quasiregular curves proves normality and resolves nodal points.

New proof linking scalar curvature to volume growth on 3-manifolds.

We study asset price bubbles in market models with proportional transaction costs $λ\in (0,1)$ and finite time horizon $T$ in the setting of [49]. By following [28], we define the fundamental value $F$ of a risky asset $S$ as the price of a super-replicating portfolio for a position terminating in one unit of the asset…

In a recent comment (Johansen A 2003 An alternative view, Quant. Finance 3: C6-C7, cond-mat/0302141), Anders Johansen has criticized our methodology and has questioned several of our results published in [Sornette D and Zhou W-X 2002 The US 2000-2002 market descent: how much longer and deeper? Quant. Finance 2: 468-81,…

Generalized stability theorem for compact manifolds with boundary.

Study on prescribing positive curvature with conical singularities on a sphere.

The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.

Study asset price bubbles in markets with short sales prohibitions and model uncertainty.

Proves effective linear volume growth for 3-manifolds with positive scalar curvature.

Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.

The paper improves stability estimates for soap bubble theorem in curved domains.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

Paper proves new inequalities for Einstein-Maxwell data sets.

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single stand…

New positive mass theorems for ALH manifolds with toroidal ends.

The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.