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 NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. SCHWARZ’S LEMMA FROM A DIFFERENTIAL GEOMETRIC VIEWPOINT IISc Lecture Notes Series — Vol. 2 Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. 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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 September9,2010 18:12 WorldScienti(cid:12)cBook-9inx6in schwarzs To our mothers v This page is intentionally left blank October13,2010 10:29 WorldScientificBook-9inx6in series˙preface Series Preface WorldScientificPublishingCompany-IndianInstituteofScienceCollaboration IIScPressandWSPCareco-publishingbooksauthored byworld renownedsci- entistsandengineers. Thiscollaboration,startedin2008duringIISc’scentenary year under a Memorandum of Understanding between IISc and WSPC, has re- sultedintheestablishmentofthreeSeries:IIScCentenaryLecturesSeries(ICLS), IIScResearchMonographsSeries(IRMS),andIIScLectureNotesSeries(ILNS). This pioneering collaboration will contribute significantly in disseminating currentIndianscientificadvancementworldwide. The “IISc Centenary Lectures Series” will comprise lectures by designa- ted Centenary Lecturers - eminent teachers and researchers from all over the world. The “IISc ResearchMonographsSeries” will comprise state-of-the-art mono- graphswrittenbyexpertsinspecificareas. Theywillinclude,butnotlimitedto, theauthors’ownresearchwork. The“IIScLectureNotesSeries”willconsistofbooksthatarereasonablyself- containedandcanbeusedeitherastextbooksorforself-studyatthepostgraduate level in science and engineering. The books will be based on material that has beenclass-testedformostpart. EditorialBoardfortheIIScLectureNotesSeries(ILNS): GadadharMisra,Editor-in-Chief([email protected]) ChandrashekarSJog([email protected]) JoyKuri([email protected]) KLSebastian([email protected]) DiptimanSen([email protected]) SandhyaVisweswariah([email protected]) vii This page is intentionally left blank 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 ix