Tohoku Mathematical Journal
2011
March
SECOND SERIES VOL. 63, NO. 1

Tohoku Math. J.
63 (2011), 41-57

Title GEOMETRIC INVARIANTS ASSOCIATED WITH PROJECTIVE STRUCTURES AND UNIVALENCE CRITERIA

Author Seong-A Kim and Toshiyuki Sugawa

(Received December 3, 2009)
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.

2000 Mathematics Subject Classification. Primary 30F45; Secondary 30C55, 53A30.
Key words and phrases. Schwarzian derivative, conformal metric, univalence criterion.

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