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Generalized Schwarzians in several variables and Möbius invariant differential operators

Generalized Schwarzians in several variables and Möbius invariant differential operators Abstract. We introduce certain Mobius invariant di¨erential operators in several variables È which generalize the classical Schwarzian derivative in one variable. We show that these generalized Schwarzians generate the entire set of Mobius invariant di¨erential operators with È function coe½cients acting on local biholomorphisms. We also give a necessary and su½cient condition to prescribe generalized Schwarzians and give the formula for the solution to the Schwarzian equation. 1991 Mathematics Subject Classi®cation: 30. Contents 1. Introduction and summary of results 2. Canonical lifts of locally biholomorphic maps 3. Generalized Schwarzians 4. Generalized Schwarzians as di¨erential operators 5. Generating Mobius invariant di¨erential operators È 6. Prescribing generalized Schwarzians References 165 167 169 175 178 181 188 §1 Introduction and summary of results The classical Schwarzian derivative, S f , of a holomorphic function of one complex variable with nonvanishing derivative on a domain r C is given by 2 H 1 f HH z 1 f HH z X 1-1 S f z À 2 f H z 4 f H z Here, we deliberately chose the coe½cient of the ®rst term to be À1a2, and accordingly the coe½cient of the second term to be 1a4 since various formulae turn http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Generalized Schwarzians in several variables and Möbius invariant differential operators

Molzon, Robert; Tamanoi, Hirotaka
Forum Mathematicum , Volume 14 (2) – Jan 29, 2002
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References (7)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.008
Publisher site
See Article on Publisher Site

Abstract

Abstract. We introduce certain Mobius invariant di¨erential operators in several variables È which generalize the classical Schwarzian derivative in one variable. We show that these generalized Schwarzians generate the entire set of Mobius invariant di¨erential operators with È function coe½cients acting on local biholomorphisms. We also give a necessary and su½cient condition to prescribe generalized Schwarzians and give the formula for the solution to the Schwarzian equation. 1991 Mathematics Subject Classi®cation: 30. Contents 1. Introduction and summary of results 2. Canonical lifts of locally biholomorphic maps 3. Generalized Schwarzians 4. Generalized Schwarzians as di¨erential operators 5. Generating Mobius invariant di¨erential operators È 6. Prescribing generalized Schwarzians References 165 167 169 175 178 181 188 §1 Introduction and summary of results The classical Schwarzian derivative, S f , of a holomorphic function of one complex variable with nonvanishing derivative on a domain r C is given by 2 H 1 f HH z 1 f HH z X 1-1 S f z À 2 f H z 4 f H z Here, we deliberately chose the coe½cient of the ®rst term to be À1a2, and accordingly the coe½cient of the second term to be 1a4 since various formulae turn

Journal

Forum Mathematicum – de Gruyter

Published: Jan 29, 2002

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