Dai–Li's energy-maximizing Hitchin section conjecture

Let $\Sigma$ be a Riemann surface and let $\mathcal M_{\text{Higgs}}(\Sigma)$ be the moduli space of Higgs bundles, equipped with its Hitchin fibration. A Hitchin section is the section of this fibration selecting the Hitchin-base point corresponding to the principal, or Fuchsian, Higgs bundle. Dai–Li's conjecture. Inside each Hitchin fiber of the moduli space $\mathcal M_{\text{Higgs}}(\Sigma)$, the Hitchin section maximizes the energy density of the corresponding harmonic maps. Theorem 1.1 proves the conjecture for all $n$-Fuchsian fibers; earlier results establish it for $n=2$ and for cyclic $SL(n,\mathbb R)$-Higgs bundles with $n=3,4$, while the general case remains open.