Tohoku Mathematical Journal

Geometric invariants associated with projective structures and univalence criteria

Seong-A Kim and Toshiyuki Sugawa

Full-text: Open access

Abstract

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the “Schwarzian derivative” of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.

Article information

Source
Tohoku Math. J. (2), Volume 63, Number 1 (2011), 41-57.

Dates
First available in Project Euclid: 19 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1303219935

Digital Object Identifier
doi:10.2748/tmj/1303219935

Mathematical Reviews number (MathSciNet)
MR2788775

Zentralblatt MATH identifier
1218.30124

Subjects
Primary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Secondary: 30C55: General theory of univalent and multivalent functions 53A30: Conformal differential geometry

Keywords
Schwarzian derivative conformal metric univalence criterion

Citation

Kim, Seong-A; Sugawa, Toshiyuki. Geometric invariants associated with projective structures and univalence criteria. Tohoku Math. J. (2) 63 (2011), no. 1, 41--57. doi:10.2748/tmj/1303219935. https://projecteuclid.org/euclid.tmj/1303219935


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