Bounded differentials on 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, 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 R³, maximal surfaces in H 2, n H^2, n and J J -holomorphic curves in H 4, 2 H^4, 2. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.