Bounded differentials on unit disk and the associated geometry

2022/09/03 by Dai, Song, Li, Qiongling · 2 citations
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.01384

Abstract

For a harmonic diffeomorphism between the Poincaré 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 ℝ³, maximal surfaces in ℍ^2,n and J-holomorphic curves in ℍ^4,2 respectively. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods.

Cited by

• Shannon entropy for harmonic metrics on cyclic Higgs bundles II
• Complete harmonic metrics and subharmonic functions on the unit disc

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