132 problems

Exact dimension formula. The dimension is

Ehrenpreis conjecture. There are such finite covers for which the Teichmüller distance between $M_\epsilon$ and $N_\epsilon$ in the suitable moduli space is less than $\epsilon$.

Nevanlinna's conjecture. The surface $\frak{M}$ is parabolic when the mean excess of its line complex is zero and hyperbolic when the mean excess is negative.

Ehrenpreis–Siegel conjecture. One can choose finite unbranched coverings $\pi_1$ and $\pi_2$ such that $\widetilde X_1$ and $\widetilde X_2$ have the same genus and there is a…

A Riemann surface is a surface equipped with a complex structure. A classical Schottky uniformization is a Schottky uniformization for which there exists a fundamental domain in…

Let a compact $m$-sheeted Riemann surface be partitioned into sheets by the Nuttall partition associated with a highlighted point. Denote the topmost closed sheet by the $m$th shee…

Meromorphic invariant potential conjecture. Every constant mean curvature surface in $\mathbb{R}^3$ associated with $M$ can be obtained from an invariant meromorphic potential.

Let $M$ be an open Riemann surface, that is, a connected noncompact Riemann surface without boundary. A holomorphic embedding $X\colon M\to\mathbb{C}^2$ is proper if $X^{-1}(K)$ is…

D'Hoker–Phong's conjecture. The genus-two formula could be extended to genus $g>2$ by

Let ${\mathcal M}_g$ be the moduli space of Riemann surfaces of genus $g\geq 2$, and let the Weil–Petersson metric be the Kähler metric on ${\mathcal M}_g$. Its Ricci curvature is…

Uniform lower-bound conjecture. There is a number $C(m,g)>0$, depending only on $m$ and $g$, such that

Let $H$ be a finite subgroup of ${\rm Mod}_g$, the mapping class group of a closed genus-$g$ surface. Nielsen Realization conjecture. The group $H$ acts as a group of automorphisms…

Let the extremal genus two curve from the $D_3$ family be the curve discussed in the paper, and let $g\geq 3$ be an integer. Higher-genus extremal-curve conjecture. In genera $3$ a…

Let $\Pi:{\cal{P}}\to{\Bbb S}^2$ be a degree-two branched covering ramified at six points, and let ${\cal{P}}$ be the resulting Bolza surface. Let $g_0$ be the pullback of the roun…

Let $\Sigma$ be a fixed compact surface of genus greater than $1$, let $\mathcal{T}(\Sigma)$ be its Teichmüller space, and let $d_{\mathcal{T}(\Sigma)}$ be the Teichmüller metric.…

Ehrenpreis conjecture. The baseleaf preserving mapping class group has dense orbits in the Teichmüller space of the solenoid.

Area filling conjecture. There is a point $p\in S$ with

Let an annulus have analytically parametrized boundary components. Cutting a hole in it, analytically parametrizing the new boundary component, and sewing an analytically parametri…

Let $U\subset\mathbb{D}$ be a domain obtained by removing a finite or countable family of pairwise disjoint closed disks $D_i$. Let $r_i$ be the radius of $D_i$ with respect to the…

Let an $N\times N$ matrix Riemann–Hilbert problem have quasi-permutation monodromies, and let its corresponding tau-function and the appropriately defined determinant of a Cauchy–R…

Let $M$ be a Riemann surface of genus $g>1$, and let $M_N$ denote the moduli space of $N$ vortices on $M$. Its volume is defined by … where $\omega$ is the symplectic form on $M_N$…

Soliton–surface correspondence conjecture. The relation above is true; in particular, the soliton describes the "unoriented" pretzel with $g=2$.

Novikov's conjecture. The function $u(x,y,t)$ satisfies the KP equation if and only if $\Omega$ is the period matrix of some compact Riemann surface $\Sigma$.

Let $\bar{M}$ be the curve under consideration, and let $\cal{L}$ and $\cal{M}$ be two bundles on $\bar{M}$ given by gluing conditions. Log-convexity conjecture. If both…

Let $M$ be a complex curve equipped with two families of cuts satisfying the conditions in the preceding conjecture. For each integer $n\geq0$, let the associated set of $C^n$-poin…