Bounded differentials on the unit disk and the associated geometry (Q6571608)

A Higgs bundle on a Riemann surface \(\Sigma\) consists of a pair: a holomorphic vector bundle \(E\) over \(\Sigma\) and a Higgs field, represented as an \(\mathrm{End}(E)\)-valued holomorphic \(1\)-form. A Hermitian metric \(h\) on \(E\) is called harmonic if it gives rise to an equivariant harmonic map from the universal cover of \(\Sigma\) to the symmetric space \(\mathrm{GL}(r,\mathbb{C})/U(r)\), and thus is a solution to Hitchin equations. Similar to the case of a compact surface, one can define the Hitchin section consisting of Higgs bundles over the disk \(\mathbb{D}\), denoted as \(s(q_{2}, \dots, q_{r})\), parametrized by a tuple of differentials.\N\NThis paper focuses on cyclic (only \(q_{r} \neq 0\)) and subcyclic (only \(q_{r-1}\neq 0\)) Higgs bundles in the Hitchin sections over \(\mathbb{D}\) and investigates the relationship between bounded \(r\)-differentials or \((r-1)\)-differentials with the geometry of harmonic maps. In [``Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces'', Preprint, \url{arXiv:2010.05401}], the second author and \textit{T. Mochizuki} showed that there exists a unique strongly complete solution to Hitchin equations on the cyclic rank \(r\)-Higgs bundle \(s(0, \cdots, 0, q)\). The authors introduce a similar notion of strongly complete solution to Hitchin equations for subcyclic rank \((r-1)\)-Higgs bundles \(s(0, \cdots, 0,q,0)\) and show that if \(f:\mathbb{D} \rightarrow \Sigma \subset \mathrm{SL}(r,\mathbb{C})/U(r)\) is a minimal immersion induced by a holomorphic \(r\)-differential \(q\) for \(r\geq 3\) or a holomorphic \((r-1)\)-differential \(q\) for \(r \geq 4\) arising from the strongly complete solution, then the following are equivalent:\N\begin{itemize}\N\item[1.] \(q\) is bounded with respect to the hyperbolic metric \(g_{\mathbb{D}}\) on \(\mathbb{D}\);\N\item[2.] The induced metric of \(\Sigma\) is mutually bounded with \(g_{\mathbb{D}}\);\N\item[3.] There exists a constant \(\delta>0\) such that the curvature \(K\) of every tangent plane to \(\Sigma\) satisfies \(K\leq -\delta\);\N\item[4.] The induced curvature on \(\Sigma\) is bounded from above by a negative constant.\N\end{itemize}\N\NAs an application, they prove the equivalence between the boundedness of holomorphic differentials and having a negative upper bound for the induced curvature on hyperbolic affine spheres in \(\mathbb{R}^{3}\), maximal surfaces in \(\mathbb{H}^{2,n}\) and \(J\)-holomorphic curves in \(\mathbb{H}^{4,2}\).