9 problems

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation i…

Let $(\tilde{E},\tilde{\phi})$ be a Higgs bundle in the Hitchin component, and let $(E,\phi)$ be a distinct polystable $SL(n,\mathbb{C})$-Higgs bundle in the same Hitchin fiber. Le…

Let $\mathcal{M}_{Higgs}$ be the moduli space of $SL(n,\mathbb{C})$-Higgs bundles, with the action … of $\mathbb{C}^*$. For a corresponding harmonic map, write $e(f)$ for its energ…

Labourie's conjecture. There is a unique complex structure $J\in\Teich(\Sigma)$ on $\Sigma$ such that the $\rho$-equivariant harmonic map from $(\widetilde{\Sigma},J)$ to $G/K$ is…

Negative curvature conjecture. The minimal immersion is never tangential to any flat inside the symmetric space. Consequently, the sectional curvature of the immersed minimal surfa…

Modified Lee–Zhang conjecture.

Lee–Zhang conjecture. For any Hitchin representation $\rho:\Gamma\to\operatorname{PSL}(n,\mathbb{R})$, there is a Fuchsian representation $j$ such that

Let $S$ be the surface and $\rho$ a representation in $Hit_n(S)$. A representation $\rho'$ is in the Fuchsian locus of $Hit_n(S)$ when it is a Fuchsian representation viewed in…

Embedding conjecture. The map $f_{\rho}$ is an embedding.