3 problems

Densest-arrangement conjecture. The ball arrangement $\mathcal{B}_{\mathrm{opt}}(R_4,K_4)$ provides the densest congruent geodesic ball packing for the Thurston geometries.

Consider the isoperimetric problem in $\operatorname{Nil}_3$ and in the universal cover of $\operatorname{PSL}(2,\mathbb{R})$. The canonical rotational spheres are the rotational C…

Let $\mathbb{E}^3(\kappa,\tau)$ denote the homogeneous $3$-spaces in the Berger-sphere, Heisenberg and universal-cover cases with $\tau\ne 0$, including $\operatorname{Nil}_3$, the…