14 problems

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 $(P,\Phi)$ 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 $V_{(J,A)}$ 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 $h_{\mathbf{H}^2}$ 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…

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 $\Sigma$ 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 $G^\tau$ be a split real simple Lie group, and let $\theta$ be a representation in the Hitchin component. For a conformal structure on $\Sigma$, let…