论文标题
单位磁盘和相关几何形状上的有界差异
Bounded differentials on unit disk and the associated geometry
论文作者
论文摘要
对于庞加莱磁盘之间的谐波差异性，WAN显示了Hopf差异和准符合性格的界限之间的等效性。在本文中，我们将将这一结果从二次差异概括为$ r $ divferetials。我们研究了有界的全态$ r $ difterentials与从单元磁盘到对称空间$ SL（R，\ Mathbb r）/SO（R）$的相关谐波图的引起曲率之间的关系。此外，我们还显示了全体形态差异的界限与在$ \ Mathbb {r}^3 $中具有诱导的曲率的负上限之间的等效性$ \ mathbb {h}^{4,2} $。 Benoist-Hulin和Labourie-Toulisse先前已使用不同的方法获得了其中一些等价。
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 $\mathbb{R}^3$, 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.
