A study on holomorphic mappings and harmonic mappings on the unit balls

点击翻译按钮获取中文摘要

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds, and as a consequence, we obtain a univalent solution for any Loewner partial differential equation. We proved that the Loewner variation of an extreme point (respectively a support point) is also an extreme point (respectively a support point). We proved the two-point distortion theorems for affine and linearly invariant family of harmonic functions on the unit disc or of pluriharmonic mappings on the Euclidean unit ball. We obtained upper bounds for the distortion and the growth of holomorphic linearly invariant families on homogeneous unit balls.