Bounded differentials on the unit disk and the associated geometry

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] 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 r -differentials. We study the relationship between bounded holomorphic r r -differentials/ ( r − 1 ) (r-1) -differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space S L ( r , R ) / S O ( r ) SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in R 3 \mathbb {R}^3 , maximal surfaces in H 2 , n \mathbb {H}^{2,n} and J J -holomorphic curves in H 4 , 2 \mathbb {H}^{4,2} . Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.