5 problems

Discrete uniformization conjecture. Suppose that $S$ is a genus zero complete hyperbolic surface with countably many ends, all but at most one of which are cusp ends. Then $S$ is i…

Let $S$ be a compact piece of a smooth surface in $\mathbb{R}^n$, and let $N_\delta(M)$ denote its $\delta$-neighborhood. A family of parallelograms has width at least $\delta$ if…

Let $K$ be a fibered knot in a manifold $M$, and let $\xi$ be a contact structure on $M$. Non-thickenable-tori conjecture. The knot type $K$ admits non-thickenable tori in $\xi$ if…

Circle-type convex-hull uniqueness conjecture. If these two boundaries are isometric, then $X$ and $Y$ differ by a Mf6bius transformation.

Circle-type realization conjecture. There exists a circle type closed set $X\subset\mathbb{C}\cup\{\infty\}$ such that $\Omega$ is isometric to the boundary of the convex hull of…