4 problems

Hauswirth's three-phase conjecture. The isoperimetric profile of $T^2\times\mathbb R$ is composed of three parts: for small volumes, isoperimetric regions are spheres $B^3_R$; for…

Transition-volume equality conjecture. In fact,

Three-phase profile conjecture. The precise isoperimetric profile of $(T^2\times {\mathbb R},h_2+dt^2)$ should be generated by regions such as spheres $B^3_R$, cylinders…

Let $M^{n_1}_1$ and $M^{n_2}_2$ be Riemannian manifolds. Suppose that $M^{n_1}_1$ can be locally isometrically immersed in $\mathbb{R}^{n_1+p_1}$ but in no lower-dimensional Euclid…