19 problems

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…

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…

Non-tangency conjecture. Harmonic maps for Hitchin representations to $\operatorname{SL}(n,\mathbb R)$, at immersed points, are never tangent to flats in the symmetric space.

Hitchin-fiber domination conjecture. In each fiber of the Hitchin fibration, the Hitchin representation dominates every other representation.

Let $G$ be a semisimple real Lie group of real rank $r$ with finite center and no compact factors, and let Hitchin rays be rays in the representation or Higgs-bundle parameter spac…

Let $S$ be a closed surface, let $\mathrm{Hit}_{2n}(S)$ and $\mathrm{QHit}_{2n}(S)$ denote the Hitchin and quasi-Hitchin representation spaces, respectively, and consider the geome…

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 $S$ be a closed orientable surface, let $\widehat{\mathcal H}_3(S)$ be the augmented Hitchin component, and let $d^\phi$ be the path pseudo-metric on its strata induced by the…

Let $S$ be a closed orientable surface, let $d\ge 3$, and let $\mathcal H_d(S)$ be the Hitchin component with pressure metric $\mathbb P^{\omega_1}$. Let the Fuchsian locus be the…

Let $S_{g,k}$ be a punctured surface with fundamental group $Pi$, and let $n>2$. Consider a generic representation $\rho:\Pi\to {\mathrm{PSL}_n(\mathbb{C})}$ in a relative represen…

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…

Let $V$ be a real vector space and let $\Gamma$ be the surface group under consideration, with Gromov boundary $\partial\Gamma$. Let $\mathcal{F}(V)$ denote the flag variety. A con…

Simple Hilbert length rigidity conjecture. If $\rho,\sigma\in{\mathcal H}_{d}(S)$ have the same marked simple Hilbert length spectrum, then they either agree or differ by the contr…

Metric domination conjecture. The pullback metric satisfies

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.