Related papers: Double bubbles in $S^3$ and $H^3$

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…

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 least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble.

Sullivan's multi-bubble isoperimetric conjectures in $n$-dimensional Euclidean and spherical spaces assert that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$.…

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…

We characterize the critical points of the double bubble problem in $\mathbb{R}^n$ and the triple bubble problem in $\mathbb{R}^3$, in the case the bubbles are convex.

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…

The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the…

Using Brakke's Evolver, we numerically verify previous conjectures for optimal double bubbles for density $r^p$ in $R^3$ and our own new conjectures for triple bubbles.

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…

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…

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques…

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.…

We show that each central configuration in the three-dimensional hyperbolic sphere is equivalent to one central configuration on a particular two- dimensional hyperbolic sphere. However, there exist both special and ordinary central…

We construct a pure two-bubble solution for the focusing, energy-critical Hartree equation in space dimension $N \geq 7$. The constructed solution is spherically symmetric, global in (at least) the negative time direction and asymptotically…

The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two…

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.

The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions…

In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…

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…