Tohoku Mathematical Journal

Geometric invariants associated with projective structures and univalence criteria

Seong-A Kim and Toshiyuki Sugawa

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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

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

First available in Project Euclid: 19 April 2011

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Zentralblatt MATH identifier

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

Schwarzian derivative conformal metric univalence criterion


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.

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