62 problems

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.…

Let $\Sigma$ and $\Sigma'$ be closed Riemann surfaces of genus greater than $1$, regarded as Riemannian manifolds with metrics of curvature $-1$. The Ehrenpreis conjecture concerns…

Let $S_1$ and $S_2$ be closed Riemann surfaces of the same genus, and let $\tilde S_i$ denote covers of $S_i$ for $i=1,2$. A cover is conformal when the covering map is conformal,…

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…

Let $M$ be a complex curve. Let $\gamma_{i}^{\left(1\right)}$, $i\in I^{\left(1\right)}$, be a disjoint family of smooth embedded cycles in $M$ such that…

Let a Kleinian group be called admissible if it has generators $\varphi_i$, $i\in I_+$, satisfying the conditions of the amplification. For an admissible Kleinian group, a fundamen…

Exact dimension formula. The dimension is

Bolza–Klein bass-note conjecture. (a) The Bolza surface uniquely maximizes the bass note among closed and orientable hyperbolic surfaces. (b) The Klein quartic uniquely maximizes t…

Eventual 8g conjecture.

The discussion concerns three local balance conditions: the condition for Speiser graphs in Theorem, the condition in section S, and the condition in section S. Equivalence conject…

Bergman density rigidity conjecture. Then $\Phi$ is either holomorphic and

Let $p\geqslant 2$ be a prime number and let $(S,N)$ be a $\mathbb{Z}_p^m$-action of signature $(0;p^{n+1})$, where $2\leqslant m\leqslant n-1$. For each subgroup $L\leqslant N$ wi…

For a set of minima, let $C_{\mathrm{max}}$ be the largest set of curves containing $C$ for which every nonfilling subset $C'$ of $C_{\mathrm{max}}$ gives…

Arbitrary-genus symplectic-leaf conjecture. For every integer $k\in\mathbb{N}$, an appropriate choice of $U$ produces a sample of symplectic leaves that are topologically Riemann s…

Analytic Bergman-space conjecture. The statement of the smooth affine algebraic-curve theorem holds for $M$: for every open subset $U\subseteq M$, the Bergman space $A^2(U,\sigma)$…

Let $\pi$ be the projection from the marked cycle curve $\operatorname{Cyc}_p(\mathcal{F}_1)$ to its parameter space, and consider the lift of the real axis under this projection.…

Rank conjecture. The rank of the complex derivative of $\rho$ at $(i_1,i_2)$ satisfies:

Existence conjecture. There exists a strictly polystable extension of $L^{-1}$ by $L$. This would extend the existence results for reducible cone spherical metrics to generic line…

Maldacena's conjecture. Every connected Riemann surface of genus at least $2$ is the asymptotic boundary of a Schottky manifold with negative renormalized volume.

Generic transversality conjecture. The two real submanifolds $\mathrm{Op}_P(\mathscr{X},G)$ and $\mathrm{Rep}_P^0(\Gamma,G(\mathbb{R}))$ intersect transversally at every point. Equ…

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_\Omega$, let $u$ be harmonic on $\Omega$, and set $\rho=e^{-2u}$. Define $B_{\Omega,\rho}$ from an…

Buser's conjecture. The surface $M$ has a pants decomposition in which each curve has length at most

Let $f\in\mathbb C(z,w)$ and $f_\infty\in\mathscr H(\infty)$. Assume the setting of the subsection in which $E\subset\mathbb R$, $E\cap F=\varnothing$, and $F$ is symmetric with re…

Uniqueness conjecture. If $\hat{\Phi}$ is holomorphic, then $\hat{h}=\operatorname{identity}$.