4 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 $G$ and $\hat{G}$ be a pair of simple, 3-connected plane graphs corresponding to a cellular decomposition of $\mathbb{C}$ and its dual. A circle configuration is a configuratio…

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$, $\chi(M)=0$, and $d$ is a piecewise linear metric on…

Let $f\in C_{\hat{\alpha},\hat{\eta}}$ be a flat label, where $C_{\hat{\alpha},\hat{\eta}}$ denotes the space of labels with the prescribed data $\hat{\alpha}$ and $\hat{\eta}$. A…