28 problems

Labourie's conjecture. The quotient of $\mathcal{T}_H(S,\operatorname{PSL}(n,\mathbf{R}))$ by the mapping class group is a holomorphic vector bundle over $\mathcal{M}(S)$, with fib…

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…

Reid's conjecture. $$ -Finsler metrics are determined by their length spectra, and these length spectra form a dense subset of

Let $(P,\Phi)$ be a $G$-Fock bundle over $S$ with hermitian structure $\rho$ such that $\Phi\in\mathcal{P}$ is positive. Let $\mu$ be the induced $\mathfrak{g}$-complex structure o…

Let $(P,\Phi_0,\sigma)$ be a $G$-Fock bundle over $S$ equipped with a compatible hermitian structure $\rho$. Let $\eta\in\Omega^0(S,\mathfrak{g}_P^{\sigma,-\rho})$ be a hermitian e…

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle. A hermitian structure $\rho$ is compatible when it commutes with $\sigma$, and positive when the associated positivity condition for the…

Let $(P,\Phi,\sigma)$ be a $G$-Fock bundle equipped with a compatible hermitian structure $\rho$. Let $d_A$ be the unique unitary, $\sigma$-invariant connection satisfying…

The geometric Hitchin space is denoted by $\hat{\mathcal{T}}^n(S)$, and the Hitchin component is the corresponding component of the character variety for a split real semisimple Li…

Let $V_{(J,A)}$ be the tangent space described by the relevant system of PDEs, and let $\mathcal{L}_{L^2}$ be the space of $L^2$-regularity tensors representing the infinitesimal d…

Let $\Sigma$ be a smooth surface and let $\mathcal{T}^n$ be the moduli space of higher complex structures of order $n$ on $\Sigma$. Consider the cotangent bundle $T^*\mathcal{T}^n$…

Let $\Sigma$ be a smooth surface, let $n$ be a positive integer, and let $\mathcal{T}^n$ denote the moduli space of higher complex structures of order $n$ on $\Sigma$. Let the Hitc…

Labourie's conjecture. The map $\Psi$ is a diffeomorphism. This conjecture concerns a canonical parametrization of the Hitchin component by a Riemann surface together with higher h…

Let $S$ be a closed oriented surface, let $V$ be the vector bundle constructed from a covector in $T^*\mathcal{T}^n(S)$, and let…

Let $h_{\mathbf{H}^2}$ be the Hitchin map and let $\sigma_{\mathbf{H}^2}$ be the Hitchin–Li–Mochizuki section, whose image parametrizes quasisymmetric maps associated with the corr…

Let $X_0$ be a hyperbolic structure, and let $(\rho_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be rays associated to two different $d$-th order holomorphic differentials $q_1$ and…

Let $\Sigma$ be the underlying surface, let $\mathfrak{g}$ be a real split Lie algebra, let $G$ be the real split Lie group associated to $\mathfrak{g}$, and let…

Let $S$ be a closed oriented surface with genus $g\geq 2$ and $n\geq 4$. For any point $\sigma\in \mathcal{T}(S) \subset \mathcal{H}_{n}(S)$, let $X$ be the Riemann surface corresp…

Let $\Sigma$ be a closed surface of genus $g\geq 2$, and let $\hat{\mathcal{T}}^n_{\Sigma}$ denote geometric Hitchin space and $\mathcal{T}^n_{\Sigma}$ Hitchin's component. Canonic…

Let $C$ be a compact Riemann surface, and let the $n$-th Hitchin component be the connected component of … containing symmetric powers of Fuchsian representations. For a representa…

Unique minimal-surface conformal structure conjecture. For every representation in the Hitchin component there exists a unique conformal structure on $\Sigma$ whose associated Hopf…

Hitchin's conjecture. The space $\mathcal{H}_3$ is the space of conjugacy classes of monodromy representations of flat properly convex projective structures.

Kähler metric conjecture. The Hitchin component admits a mapping class group-invariant Kähler metric.

Labourie–Goldman–Wentworth conjecture. The map $\mathcal H$ is a homeomorphism for general $G$. This extends the known case $G={\bf PSL}(2,\mathbb{R})$ and…

Let $\Sigma$ be a surface, let $G$ be a split real simple Lie group, and let $\theta$ be a representation in a Hitchin component. Let $E_\theta$ be the associated energy function o…

Let $\Sigma$ be a surface, let $G$ be a reductive Lie group, and let $\theta$ be a representation in the Hitchin component. For each conformal structure on $\Sigma$, let $\psi$ be…