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PreprintCC BY 4.0arXiv

The Schwarzian derivative in several complex variables univalence and geometry

  • Rodrigo Hernández
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Abstract

In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors--Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in ℂn. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors--Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.

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Cite

BibTeXRIS
BibTeX
@misc{hernandez2026schwarzian,
  title = {{The Schwarzian derivative in several complex variables univalence and geometry}},
  author = {Rodrigo Hernández},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09133},
  eprint = {2610.09133},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09133},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Schwarzian derivative in several complex variables univalence and geometry
AU  - Rodrigo Hernández
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.09133
UR  - https://arxiv.org/abs/2610.09133
AB  - In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors--Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in ℂ^n. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors--Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.
ER  -