arXiv research

Survey on soap bubble partitions and their stability.

The paper disproves compactness for high-energy Willmore immersions and finds minimal bubbles on Willmore surfaces.

Paper proves convergence for Willmore immersions with minimal bubbles.

The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…

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.

Hexagonal norm double bubble problem solved with minimal configurations.

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

Removes singularity order for Willmore immersions, reducing bubbling scenarios.

Study of immersions with Willmore energy leading to spherical and catenoid bubbles.

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,…

Solves the quintuple bubble problem on spheres and Euclidean spaces.

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

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 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…

Classifies low energy maps from curved surfaces into spheres.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …

New proof of Bishop's theorem using soap bubbles with singularities.

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show th…

Study shows price bubbles can exist even with heterogeneous beliefs.

Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.

Modeling financial bubbles and crashes with a cubic momentum function.

This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.

Study detects Chinese stock market bubbles using LPPLS confidence indicator.

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…

A new model detects financial bubbles with high accuracy.

We develop a theory of axisymmetric surfaces minimizing a combination of surface tension and nematic elastic energies which may be suitable for describing simple film and bubble shapes. As a function of the elastic constant and the applied tension on the bubbles, we find the analogues of the unduloid, sphere, and nodoi…

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.

Study on membranes under confinement, proving existence and regularity of minimizers.

Proves existence and regularity of spherical minimizers for lipid membrane energy.

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 $120^{\circ}$ …

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…

The paper studies solutions to a $Q$-curvature equation on 4-manifolds with bubbling phenomena.

Study finds small surfaces in space times with new functionals.

Researchers define and prove existence of minimizers for generalized Willmore functionals.

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

Paper proves no specific CMC hypersurfaces in hyperbolic space.

We apply Geometric Arbitrage Theory to obtain results in mathematical finance for credit markets, which do not need stochastic differential geometry in their formulation. We obtain closed form equations involving default intensities and loss given defaults characterizing the no-free-lunch-with-vanishing-risk condition …

New proof of Riemannian Penrose inequality in 3D without horizons.

Characterizes critical points in convex double and triple bubbles.

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 appli…

The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…

Standard bubbles and partitions are stable in various model spaces.

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

Clusters on simple manifolds have connected boundaries.