142 problems

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 $\epsilon$.

The universal Teichmüller space is the space of boundary values of quasiconformal self-maps of the disk; equivalently, it can be viewed through quasisymmetric self-maps of…

Thurston's conjecture. This theorem should have a simpler proof based on more general principles, in particular the max flow/min cut principle, convexity, and…

Kojima–Mizushima–Tan conjecture. The map

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…

Let $S$ be a closed, oriented surface of genus at least $2$, let $\Gamma$ be the 1-skeleton of a triangulation of $S$, let $\mathcal C_\Gamma$ be the space of pairs $(\sigma,C)$ co…

Let $f\in\Sigma(0)$ be a univalent function with a quasiconformal extension, let $k(f)$ denote its Teichmüller norm, and let $\varkappa_p(f)$ denote its $p$-th truncated Grunsky no…

Let $\Sigma_{g,n}$ be a punctured surface, let $\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}$ be its complexified curve algebra, and let $\mathcal{T}^{d}(\Sigma_{g,n})$ be its decorated…

Let $\mathcal{T}:=\mathcal{T}_{g,n}$ be the Teichmüller space of a finite-type orientable surface, and let $\tau:\mathbb{H}\to\mathcal{T}$ be a Teichmüller disk generated by a holo…

Let $d\zeta^{2}$ be an everywhere finite quadratic differential on a surface $\frak{F}$, different from $0$. At each point of $\frak{F}$, consider the direction in which…

Let $S$ be a surface, let $\mathcal{T}(S)$ be its Teichmüller space, and fix a measured lamination $\lambda$ on $S$. Grafting along $\lambda$ induces a self-map of $\mathcal{T}(S)$…

Grothendieck's reconstruction conjecture. The entire Teichmüller tower can be reconstructed from the first two levels via gluing: level $1$ provides a complete system of generators…

Let $S$ be a genus-$2$ surface, and consider the Maskit embedding obtained by shrinking a dividing curve and a non-dividing curve, so that the resulting surfaces are a punctured to…

Let ${\mathcal T}^{\text{s}}_{1,1}$ denote the set of hyperbolic one-holed tori whose isometry group is isomorphic to $C_2\times S_3$. If $M\in {\mathcal T}^{\text{s}}_{1,1}$, and…

A once-punctured torus is a hyperbolic surface whose simple closed geodesics have a length spectrum, with multiplicity counting how many distinct simple closed geodesics have a giv…

Imaginary-part minimum conjecture. It is conjectured that

Luo's bounded-polytope conjecture. When $\lambda<0$, the image of $T(S)$ under the $\psi_{\lambda}$ edge-invariant coordinate is a bounded convex polytope.

Let $S$ be a closed surface, let $\mathcal E^{(n)}$ be the vector bundle over Teichmüller space whose fibre at a complex structure $J$ is … and let … be the Hitchin map, defined by…

Let $S$ and $R$ be Riemann surfaces, let $f_1:S\to R$ be a quasiconformal map, and let $\mathcal{CM}(R)$ be the space of conformal metrics on $R$ with unit area and at most conical…

Zero-intersection characterization. For any $F\in\mathcal{PMF}$,

Let $S$ be a closed oriented surface, let $\rho$ be a Hitchin representation, and let $e_\rho$ denote its energy functional on Teichmüller space. Nondegeneracy conjecture. The mini…

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, let $\ell$ be a base leaf, and let ${\sf M}(\widehat{\Sigma},\ell)$ be its mapping class group…

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, with Teichmüller space $\mathcal{T}(\widehat{\Sigma})$. Let ${\sf M}(\widehat{\Sigma},\ell)$ b…

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.…