74 problems

Let ${\mathbb H}^1$ be the three-dimensional sub-Riemannian Heisenberg group. A Pansu sphere is the surface obtained as the union of all sub-Riemannian geodesics of a fixed curvatu…

Maggi's conjecture. Wulff shapes are the unique such sets.

Let $B_1$ be the open unit ball in $\mathbb{R}^n$, let $G_\lambda(x)=\lambda^{n+1}G(\lambda x)$ for $\lambda>0$, and define … Consider … with $\gamma>0$. Large-parameter unit-ball…

Let $A_1$, $A_2$, and $A_3$ be positive real numbers. Consider planar graphs enclosing and separating finite areas $B_1$, $B_2$, and $B_3$ satisfying … A standard triple bubble is…

Let $A_1$, $A_2$, and $A_3$ be three finite areas. A regular triple bubble complex is a planar graph with four vertices, six edges, and three connected regions. Triple bubble conje…

For $n>2$, the $n$-bubble problem in $\mathbb{R}^2$ seeks a least-perimeter graph enclosing and separating $n$ prescribed areas. General $n$-bubble conjecture. The least perimeter…

A tiling of $\mathbb{R}^2$ separates the plane into pieces of unit area, and its perimeter is interpreted appropriately despite the infinite extent of the region. Honeycomb conject…

Let $n\ge 1$, let $k\le n+2$, and prescribe the volumes of $k$ regions in $\mathbb{R}^{n+1}$. A standard $k$-bubble is the standard soap-bubble configuration enclosing those $k$ re…

Let $n\ge 1$ and consider double bubbles in $\mathbb{R}^{n+1}$ enclosing two regions of prescribed volumes. A stable double bubble is a double bubble stable under volume-preserving…

Let two regions in $\mathbb{R}^3$ have prescribed volumes. A standard double bubble consists of two spherical caps separated by a spherical cap or a flat disc, with the three piece…

Let $A_0\geq A_2\geq A_1$ be areas, and let a flat torus have area $A_0+A_1+A_2$. A non-tiling minimizing double bubble is a perimeter-minimizing double bubble enclosing areas…

Consider the infinite Möbius strip, namely the surface between two horizontal lines in the plane, where the lines are identified with a flip. Infinite Möbius-strip conjecture. The…

Let a perimeter-minimizing double bubble be a double bubble enclosing prescribed areas with least perimeter on a flat Klein bottle. Klein bottle conjecture. Perimeter-minimizing do…

Let a minimizing graph be a least-perimeter partition of the unit disk into regions of prescribed areas. Connectedness conjecture. A minimizing graph separates the disk into connec…

Let the ambient space be either the three-dimensional sphere or the three-dimensional hyperbolic space, and consider a standard double bubble enclosing two prescribed volumes. Cott…

Small-volume triple-bubble conjecture. The minimizers will look like the double bubbles of Figure 2 with a small ball attached, and the phase diagram will look just like Figure 3.

Complete-classification conjecture. The double bubbles of Figure 1 together with the Hexagonal Honeycomb of Figure 4 comprise the complete set of area-minimizing double bubbles for…

Hexagonal Honeycomb conjecture. For these volumes, the Hexagonal Honeycomb prism of Figure 4 is a perimeter-minimizing double bubble.

Very-long-torus Double Slab conjecture. In the special case of a very long $T^3$, the Double Slab is optimal for most volumes.

Concavity conjecture. The least area to enclose and separate two given volumes in the three-torus is a concave function of the volumes.

Ritoré–Ros conjecture. The optima for the isoperimetric problem in a cubic $T^3$ are the sphere, cylinder, and slab.

Central conjecture. The ten different types of two-volume enclosures pictured in Figure 1 comprise the complete set of surface-area-minimizing double bubbles in the flat cubic thre…

Let $N$ be the number of points and let $(P_m)$ denote the property that the maximizer represented by regular polygons has the specified geometric extremal behavior for $m$ points.…

Let $\Pi_n$ be an $n$-gon with side lengths $a_1,\ldots,a_n$, perimeter $L_n$, and area $A_n$. Define its pseudo-area by … For the regular $n$-gon, let … Define the ratio…

Let the sub-Lorentzian Heisenberg group be equipped with its horizontal distribution and sub-Lorentzian metric. Consider the family of smooth, acausal, boost-symmetric surfaces wit…