Bounded differentials on unit disk and the associated geometry

TL;DR

This paper extends the relationship between bounded differentials and geometric properties from quadratic to r-differentials, linking boundedness to curvature bounds in various geometric contexts involving harmonic maps and Higgs bundles.

Contribution

It generalizes known results from quadratic differentials to r-differentials, establishing new equivalences between boundedness and curvature bounds in diverse geometric settings.

Findings

Bounded holomorphic r-differentials relate to curvature of harmonic maps.

Equivalence between boundedness of differentials and negative curvature bounds.

Connections to hyperbolic affine spheres, maximal surfaces, and J-holomorphic curves.

Abstract

For a harmonic diffeomorphism between the Poincar\'{e} disks, Wan showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to $r$-differetials. We study the relationship between bounded holomorphic $r$-differentials and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space $SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles. Also, we show the equivalences between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in $\mathbb{R}^3$, maximal surfaces in $\mathbb{H}^{2,n}$ and $J$-holomorphic curves in $\mathbb{H}^{4,2}$ respectively. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.