Schwarz's Lemma from a Differential Geometric Viewpoint

100 Pages·2010·0.676 MB·English

by  Kang-Tae Kim, Hanjin Lee

HSc Lecture Notes Series
Kang-Tac Kim
Hanjin Lee
Schwarz's Lemma from a
Differential Geometric
Viewpoint
IISc
World Scientific
Press
i
Schwarz’s Lemma from a
Differential Geometric
Viewpoint
IISc Lecture Notes Series
Schwarz’s Lemma from a
Differential Geometric
Viewpoint
Kang-Tae Kim
Pohang University of Science and Technology, Korea
Hanjin Lee
Handong Global University, Korea
World Scientific
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British Library Cataloguing-in-Publication Data
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SCHWARZ’S LEMMA FROM A DIFFERENTIAL GEOMETRIC VIEWPOINT
IISc Lecture Notes Series — Vol. 2
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4324-78-6
ISBN-10 981-4324-78-7
Printed in Singapore.
LaiFun - Schwarz's lemma from a diff.pmd 1 10/8/2010, 9:14 AM
IISc LECTURE NOTES SERIES
ISSN: 2010-2402
Editor-in-Chief:Gadadhar Misra
Editors: Chandrashekar S Jog
Joy Kuri
K L Sebastian
Diptiman Sen
Sandhya Visweswariah
Published:
Vol. 1 Introduction to Algebraic Geometry and Commutative Algebra
by Dilip P Patil & Uwe Storch
Vol. 2 Schwarz’s Lemma from a Differential Geometric Viewpoint
by Kang-Tae Kim & Hanjin Lee
LaiFun - Schwarz's lemma from a diff.pmd 2 10/8/2010, 9:14 AM
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To our mothers
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Series Preface
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October8,2010 14:42 WorldScienti(cid:12)cBook-9inx6in schwarzs
Preface
Schwarz’sLemmawasonlyasmallpreparatorylemmaatitsinitialdiscov-
ery around 1880. (It seems di(cid:14)cult to locate its historical origin precisely;
see[Osserman1999a].) Butithasgrownsomuchthatithasbecomeaprin-
cipal tool in various branches of mathematics. We were astonished when
the internet search showed more than 150 \hits" (i.e., 150 research papers
detected in the search) with the keyword \Schwarz’s Lemma". As much
as it was a surprise to us, we were inspired to work on a survey, mostly
for our own sake in the (cid:12)rst place, of the stream of research developments
pertaining to Schwarz’sLemma and its developments.
It did not take us too long to realize, after some reading, that there
are indeed quite a few but not too many fundamental achievements that
provide \core ideas and methods". Some of them, however, might not be
so easy to understand in a quick reading. Thus we were convinced that
it might be worthwhile to write these notes on its di(cid:11)erential geometric
developments in their present form.
ThesenotesareofaclassicalnatureandstartwiththeoriginalSchwarz’s
Lemma|preceded only by some preliminaries on harmonic and subhar-
monic functions (Chapter 1). The modi(cid:12)cation by Pick (around 1916; see
[Kobayashi 1970]), now known as the Schwarz-Pick Lemma, is introduced
in the same chapter and is re-interpreted in terms of the Poincar(cid:19)e metric
and distance. This establishes the beginning stage of di(cid:11)erential geomet-
ric ideas making bridges with Schwarz’sLemma. These matters constitute
Chapter 2.
In the 1930’s, the statement \Negative curvature restricts holomorphic
mappings" emerged as an important slogan in research on di(cid:11)erential ge-
ometryandgeometric(complex)analysis. ThegeneralizationsofSchwarz’s
Lemmafromtheviewpointofdi(cid:11)erentialgeometryinvolvingcurvaturehave
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