![Introduction and overview 3
measurable metric or conformal structures. Indeed, it is from this point of view
that the Beltrami equation initially arose and it is a view which we adopt in the
last few chapters, where we present a selection of topics that represent fairly
recent developments in a different direction from the analytic development in the
first part of the book.
Many of these notions, ideas and results extend to manifolds, and accordingly,
while we do not develop this aspect in full, the reader should note that all the
machinery we set up is ready for these developments. It is a deep result of Sullivan
that all topological n-manifolds (n 54 4) admit quasiconformal structures, and we
discuss this later in our overview. Thus one is able to do analysis on purely
topological objects, relating topological and analytical invariants. For instance,
Donaldson and Sullivan have developed a measurable Yang-Mills theory [78],
there are analogues of the Atiyah-Singer index theory [321], and there is also the
recent work of Connes, Sullivan and Teleman [73] developing the theory of
characteristic classes in this setting.
Quasiconformal mappings provide a class which lies between homeomorph-
isms and diffeomorphisms. Mappings of finite distortion are even more flexible.
Many constructions in analysis, geometry and topology rely on limiting processes.
The compactness properties of families of mappings with finite distortion
make them ideal tools for solving various problems in n-dimensional analysis
and topology. For instance, in studying deformations of elastic bodies and
the related extremals for variational integrals, mappings of finite distortion
are often the natural candidates to consider because they are closed under
uniform convergence, whereas the limit of a diffeomorphism need not be smooth
nor even a homeomorphism. In this book, we present a considerable number
of such compactness results for the class of mappings of finite distortion, see
Chapter 8.
In recent years there has been another well-known theory of mappings
(referred to as deformations) whose ideas have come to the core of geometry
and analysis. This is the non-linear elasticity theory of Antman, Ball and Ciarlet
[12, 21, 22, 66], building on earlier work of Green [118, 119]. The theory was
founded by the eighteenth-century mathematicians Bernoulli and Euler, who
were concerned with the practical problems of mathematics and physics of that
time. Nowadays the theory of elasticity studies mappings (in certain Sobolev
classes) which minimize stored energy integrals. These mappings are not always
quasiregular, but the governing PDEs are the same. It is necessary to study non-
linear equations to observe certain physical phenomena such as bifurcation and
phase transition. In particular, the Jacobian determinant (a highly non-linear
geometric object itself) of these mappings has been subjected to a great deal of
investigation. Its higher integrability properties were already recognized in the
celebrated paper of Gehring in the 1960s, where he discovered the "reverse Holder
inequalities". In this monograph we shall give a comprehensive account of higher
integrability properties of Jacobians and other, more general, non-linear quant-
ities which arise naturally in the L" theory of differential forms.](https://image.slidesharecdn.com/14264-250620064919-13eee1c6/85/Geometric-Function-Theory-and-Non-linear-Analysis-1st-Edition-Tadeusz-Iwaniec-25-320.jpg)
Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec
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- 5. Geometric Function Theory and Non linear Analysis 1st Edition Tadeusz Iwaniec Digital Instant Download Author(s): Tadeusz Iwaniec, Gaven Martin ISBN(s): 9780198509295, 0198509294 Edition: 1st File Details: PDF, 13.46 MB Year: 2002 Language: english
- 6. D-X-70 RD }I i i1 f 'Z S Jul e'- ra r ,itu'l, caI-, TA DE,USZ .l WAN I E C and GAVE NN Mk.R' TINT OXFOI SC +' N CE PUBUC .IONS
- 7. OXFORD MATHEMATICAL MONOGRAPHS Series Editors J. M. BALL E. M. FRIEDLANDER I. G. MACDONALD L. NIRENBERG R. PENROSE J. T. STUART
- 8. OXFORD MATHEMATICAL MONOGRAPHS A. Belleni-Moranti: Applied semigroups and evolution equations A. M. Arthurs: Complementary variational principles 2nd edition M. Rosenblum and J. Rovnyalc Hardy classes and operator theory J. W. P. Hirschfeld: Finite projective spaces of three dimensions A. Pressley and G. Segal: Loop groups D. E. Edmunds and W. D. Evans: Spectral theory and differential operators Wang Jianhua: The theory of games S. Omatu and J. H. Seinfeld: Distributed parameter systems: theory and applications J. Hilgert, K. H. Hofmann, and J. D. Lawson: Lie groups, convex cones, and semi groups S. Dineen: The Schwarz lemma S. K. Donaldson and P. B. Kronheimer: The geometry offour-manifolds D. W. Robinson: Elliptic operators and Lie groups A. G. Werschulz: The computational complexity of differential and integral equations L. Evens: Cohomology of groups G. Effinger and D. R. Hayes: Additive number theory of polynomials J. W. P. Hirschfeld and J. A. Thas: General Galois geometries P. N. Hoffman and J. F. Humphreys: Projective representations of the symmetric groups 1. Gyori and G. Lades: The oscillation theory of delay differential equations J. Heinonen, T. Kilpelainen, and O. Martio: Non-linear potential theory B. Amberg, S. Franciosi, and F. de Giovanni: Products of groups M. E. Gurtin: Thermomechanics of evolving phase boundaries in the plane 1. Ionescu and M. Sofonea: Functional and numerical methods in viscoplasticity N. Woodhouse: Geometric quantization 2nd edition U. Grenander: General pattern theory J. Faraut and A. Koranyi: Analysis on symmetric cones I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition B. L. R. Shawyer and B. B. Watson: Borel's methods of summability M. Holschneider: Wavelets. an analysis tool Jacques Thhvenaz: G-algebras and modular representation theory Hans-Joachim Baues: Homotopy type and homology P. D. D'Eath: Black holes: gravitational interactions R. Lowen: Approach spaces: the missing link in the topology-uniformity-metric triad Nguyen Dinh Cong: Topological dynamics of random dynamical systems J. W. P. Hirschfeld: Projective geometries over finite fields 2nd edition K. Matsuzald and M. Taniguchi Hyperbolic manifolds and Kleinian groups David E. Evans and Yasuyuld Kawahigashi: Quantum symmetries on operator algebras Norbert Klingen: Arithmetical similarities: prime decomposition andfinite group theory Isabelle Catto, Claude Le Bris, and Pierre-Louis Lions: The mathematical theory of thermodynamic limits: Thomas-Fermi type models D. McDuff and D. Salamon: Introduction to symplectic topology 2nd edition William M. Goldman: Complex hyperbolic geometry Charles J. Colbourn and Alexander Rosa: Triple systems V. A. Kozlov, V. G. Maz'ya and A. B. Movchan: Asymptotic analysis offields in multi-structures Girard A. Maugin: Nonlinear waves in elastic crystals George Dassios and Ralph Kleinman: Low frequency scattering Gerald W. Johnson and Michel L. Lapidus: The Feynman integral and Feynman's operational calculus W. Lay and S. Y. Slavyanov: Special functions: A unified theory based on singularities D. Joyce: Compact manifolds with special holonomy A. Carbone and S. Semmes: A graphic apology for symmetry and implicitness Johann Boos: Classical and modern methods in summability Nigel Higson and John Roe: Analytic K-homology S. Semmes: Some novel types of fractal geometry Tadeusz Iwaniec and Gaven Martin: Geometric function theory and non-linear analysis
- 9. Geometric Function Theory and Non linear Analysis TADEUSZ IWANIEC John Raymond French Distinguished Professor of Mathematics at Syracuse University and GAVEN MARTIN Professor of Mathematics at the University of Auckland and James Cook Fellow of Royal Society (NZ) CLARENDON PRESS OXFORD 2001
- 10. OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Si o Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Tadeusz Iwaniec and Gaven Martin, 2001 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2001 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data Iwaniec, Tadeusz. Geometric function theory and non-linear analysis / Tadeusz Iwaniec and Gaven Martin. (Oxford mathematical monographs) Includes bibliographical references and index. 1. Geometric function theory. 2. Non-linear theories. 3. Numerical analysis. 1. Martin, Gaven. II. Title. III. Series. QA360 .I93 2001 515-dc2l 2001034652 ISBN 0 19 850929 4 10 9 8 7 6 5 4 3 2 1 Typeset by Integra Software Services Pvt. Ltd, Pondicherry, India www.integra-india.com Printed in Great Britain on acid-free paper by T.J. International Ltd., Padstow, Cornwall
- 11. To our families Graiyna & Krystyna and Dianne & Jennifer & Amy
- 12. PREFACE This book is largely about the geometry of mappings - that is, functions or deformations between subsets of the Euclidean n-space R" and more generally between manifolds or other geometric objects. Such mappings may be homeo- morphisms, diffeomorphisms, branched coverings or more abstract cor- respondences such as Sobolev mappings. They may arise as the solutions to differential equations, the minima of certain optimization problems in the calculus of variations, as local coordinates on a manifold or as geometric realizations of abstract isomorphisms between spaces. In each case the regularity and geometric properties of these mappings will tell us something about the problem at hand or the spaces we are investigating. Of course such a general topic intersects many areas of modern mathematics. Thus we will run into aspects of differential geometry, topology, partial differen- tial equations and harmonic analysis, as well as nonlinear analysis, the calculus of variations and so forth. A good deal of this intersection is surveyed in Chapter 1, in which our aim is to give the reader some appreciation of the diversity of applications and directions in which current research is moving, as well as a glimpse of the substantial body of work which we were unable to cover in any detail here. This book is essentially a research monograph. We have tried to present a fairly complete account of the most recent developments in these areas as they pertain to the geometry of mappings, and indeed a significant portion of this book was new or recent at the time of writing. However, we do cover and offer new approaches to many aspects of the classical theory as well as devoting a few chapters to foundational material, and we have pitched the level of the book at the competent graduate student. We wish to express our deep gratitude to the many fellow mathematicians who have contributed in one way or another to this book. In particular, those from the Finnish and Italian Schools with whom we have collaborated and discussed many ideas and whose theorems can be found throughout this book. Also many thanks to Tsukasa Yashiro who created all the pictures for us, and to John Duncan and Volker Mayer who read and commented on a good portion of the manuscript. Both authors would like to acknowledge the support they received from the US National Science Foundation and the NZ Marsden Fund. While this book is dedicated to our families, there are two others we must acknowledge. These are our teachers, Bogdan Bojarski and Fred Gehring, who
- 13. Preface vii pioneered much of the theory presented here. In particular, Fred brought us together in Ann Arbor from either end of the world to do mathematics, and throughout our careers he and his wife Lois have been unfailingly supportive. Thanks! March 2001 T.I. G.M.
- 15. CONTENTS 1 Introduction and overview 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 The planar theory n-Dimensional quasiconformal mappings The Liouville theorem Higher integrability Stability and rigidity phenomena Quasiconformal structures on manifolds Nevanlinna theory Non-linear potential theory Singular integral operators Removable singularities Quasiconformal groups, semigroups and dynamics Continuum mechanics and non-linear elasticity Mostow rigidity 2 Conformal mappings 2.1 The Cauchy-Riemann system 2.2 The Mobius group 2.3 The Liouville theorem (smooth case) 2.4 Curvature 2.5 Computing the Jacobian 2.6 Conclusions 2.7 Further aspects 3 Stability of the Mobius group 3.1 Mapping classes 3.2 Harnack inequalities 3.3 A stability function 3.4 Passing Harnack inequalities on to Mt 3.5 Local injectivity 4 Sobolev theory and function spaces 4.1 Schwartz distributions 4.2 Definitions of Sobolev spaces 4.3 Mollification 4.4 Lebesgue points 4.5 Pointwise coincidence of Sobolev functions
- 16. x Contents 4.6 Alternative characterizations 4.7 Cross product of gradient fields 4.8 The adjoint differential 4.9 Subharmonic distributions 4.10 Embedding theorems 4.11 Duals and compact embeddings 4.12 Orlicz-Sobolev spaces 4.13 Hardy spaces and BMO 5 The Liouville theorem 5.1 Introduction 5.2 Second-order estimates 5.3 Identities 5.4 Second-order equations 5.5 Continuity of the Jacobian 5.6 A formula for the Jacobian 5.7 Concluding arguments 6 Mappings of finite distortion 6.1 Differentiability 6.2 Integrability of the Jacobian 6.3 Absolute continuity 6.4 Distortion functions 6.5 Examples 6.5.1 Radial stretchings 6.5.2 Winding maps 6.5.3 Cones and cylinders 6.5.4 The Zorich exponential map 6.5.5 A regularity example 6.5.6 Squeezing the Sierpinski sponge 6.5.7 Releasing the sponge 7 Continuity 7.1 Distributional Jacobians 7.2 The Ll integrability of the Jacobian 7.3 Weakly monotone functions 7.4 Oscillation in a ball 7.5 Modulus of continuity 7.6 Exponentially integrable outer distortion 7.7 Holder estimates 7.8 Fundamental LP-inequality for the Jacobian 7.8.1 A class of Orlicz functions 7.8.2 Another proof of Corollary 7.2.1
- 17. Contents xi 8 Compactness 169 8.1 Distributional Jacobians revisited 169 8.2 Weak convergence of Jacobians 172 8.3 Maximal inequalities 175 8.4 Improving the degree of integrability 176 8.5 Weak limits and orientation 181 8.6 L log L integrability 185 8.7 A limit theorem 186 8.8 Polyconvex functions 187 8.8.1 Null Lagrangians 188 8.8.2 Polyconvexity of distortion functions 190 8.9 Biting convergence 191 8.10 Lower semicontinuity of the distortion 193 8.11 The failure of lower semicontinuity 197 8.12 Bounded distortion 200 8.13 Local injectivity revisited 201 8.14 Compactness for exponentially integrable distortion 205 9 Topics from Multilinear Algebra 208 9.1 The 1-covectors 208 9.2 The wedge product 209 9.3 Orientation 211 9.4 The pullback 211 9.5 Matrix representations 212 9.6 Inner products 213 9.7 The volume element 216 9.8 Hodge duality 217 9.9 Hadamard-Schwarz inequality 220 9.10 Submultiplicity of the distortion 221 10 Differential Forms 222 10.1 Differential forms in R" 222 10.2 Pullback of differential forms 228 10.3 Integration by parts 229 10.4 Orlicz-Sobolev spaces of differential forms 232 10.5 The Hodge decomposition 234 10.6 The Hodge decomposition in R" 236 11 Beltrami equations 240 11.1 The Beltrami equation 240 11.2 A fundamental example 244 11.2.1 The construction 245 11.3 Liouville-type theorem 250
- 18. xii Contents 11.4 The principal solution 11.5 Stoilow factorization 11.6 Failure of factorization 11.7 Solutions for integrable distortion 11.8 Distortion in the exponential class 11.8.1 An example 11.8.2 Statement of results 11.9 Distortion in the subexponential class 11.9.1 An example 11.9.2 Statement of results 11.9.3 Further generalities 11.10 Preliminaries 11.10.1 Results from harmonic analysis 11.10.2 Existence for exponentially integrable distortion 11.10.3 Uniqueness 11.10.4 Critical exponents 11.10.5 Existence for subexponentially integrable distortion 11.11 Global solutions 11.12 Holomorphic dependence 11.13 Examples and non-uniqueness 11.14 Compactness 11.15 Removable singularities 11.16 Final comments 12 Riesz transforms 12.1 Singular integral operators 12.2 Fourier multipliers 12.3 Trivial extension of a scalar operator 12.4 Extension to C" 12.5 The real method of rotation 12.6 The complex method of rotation 12.7 Polarization 12.8 The tensor product of Riesz transforms 12.9 Dirac operators and the Hilbert transform on forms 12.10 The LP-norms of the Hilbert transform on forms 12.11 Further estimates 12.12 Interpolation 13 Integral estimates 13.1 Non-linear commutators 13.2 The complex method of interpolation
- 19. Contents xiii 13.3 Jacobians and wedge products revisited 343 13.4 The H'-theory of wedge products 345 13.5 An L log L inequality 347 13.6 Estimates beyond the natural exponent 350 13.7 Proof of the fundamental inequality for Jacobians 352 14 The Gehring lemma 354 14.1 A covering lemma 356 14.2 Calderdn-Zygmund decomposition 357 14.3 Gehring's lemma in Orlicz spaces 359 14.4 Caccioppoli's inequality 363 14.5 The order of zeros 367 15 The governing equations 370 15.1 Equations in the plane 370 15.2 Absolute minima of variational integrals 375 15.3 Conformal mappings 380 15.4 Equations at the level of exterior algebra 386 15.5 Even dimensions 391 15.6 Signature operators 393 15.7 Four dimensions 398 16 Topological properties of mappings of bounded distortion 401 16.1 The energy integrand 402 16.2 The Dirichlet problem 405 16.3 The A-harmonic equation 406 16.4 Caccioppoli inequality 410 16.5 The comparison principle 410 16.6 The polar set 411 16.7 Sets of zero conformal capacity 414 16.8 Qualitative analysis near polar points 416 16.9 Local injectivity of smooth mappings 419 16.10 The Jacobian is non-vanishing 422 16.11 Analytic degree theory 423 16.12 Openness and discreteness for mappings of bounded distortion 426 16.13 Further generalities 427 16.14 An update 428 17 Painlev4's theorem in space 431 17.1 Painleve's theorem in the plane 431 17.2 Hausdorff dimension and capacity 432
- 20. xiv Contents 17.3 Removability of singularities 17.4 Distortion of dimension 18 Even dimensions 18.1 The Beltrami operator 18.2 Integrability theorems in even dimensions 18.3 Mappings with exponentially integrable distortion 18.4 The Lz inverse of I -.a S 18.5 Wl"n-regularity 18.6 Singularities 18.7 An example 19 Picard and Montel theorems in space 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 Picard's theorem in space Serrin's theorem and Harnack functions Estimates in fe(Rn) Harnack inequalities near zeros Collections of Harnack functions Proof of Rickman's theorem Normal families Montel's theorem in space Further generalizations 20 Conformal structures 20.1 The space S(n) 20.2 Conformal structures 20.3 The smallest ball 21 Uniformly quasiregular mappings 21.1 A first uniqueness result 21.2 First examples 21.3 Fatou and Julia sets 21.4 Lattes-type examples 21.5 Invariant conformal structures 22 Quasiconformal groups 22.1 Convergence properties 22.2 The elementary quasiconformal groups 22.3 Non-elementary quasiconformal groups 22.4 The triple space 22.5 Conjugacy results 22.6 Hilbert-Smith conjecture 22.7 Remarks
- 21. Contents xv 23 Analytic continuation for Beltrami systems 528 23.1 Uniqueness 528 23.2 Proof of Theorem 23.1.1 529 23.3 Remarks 530 Bibliography 531 Index 547
- 23. 1 INTRODUCTION AND OVERVIEW The interplay between partial differential equations (PDEs) and the theory of mappings has a long and distinguished history, and that connection underpins this book. Gauss's practical geodesic survey work stimulated him to develop the theory of conformal transformations, for mapping figures from one surface to another. For conformal transformation from plane to plane he used a pair of equations apparently derived by d'Alembert, who first related the derivatives of the real and imaginary part of a complex function in 1746 in his work on hydro- dynamics (311, p. 497. These equations have become known as the Cauchy- Riemann equations. Gauss developed the differential geometry of surfaces around 1827, emphasizing the intrinsic geometry, with Gaussian curvature defined by measurements within the surface. If a surface is deformed conformally (preserving angles), then the Gaussian curvature is unchanged, and hence the intrinsic geometry of the surface is unaffected by such deformations. Gauss also considered geodesic curves within surfaces. In 1829 Lobachevsky constructed a surface (the horosphere) within his non-Euclidean space, such that the intrinsic geometry within that surface is Euclidean. with geodesic curves being called Euclidean lines. For the converse process, he could only suggest tentatively that, within Euclidean space, the intrinsic geometry of a sphere of imaginary radius was Lobachevskian. But imaginary numbers were then regarded with justifiable suspicion, and he did not propose that as an acceptable model of his geometry within Euclidean space. In his most famous work, Beltrami [321 showed that Lobachevsky's geometry is the intrinsic geometry of a surface of constant negative curvature, with geodesic curves being called lines in Lobachevsky's geometry. Beltrami illustrated various surfaces with constant negative curvature, the sim- plest of which is the pseudosphere generated by revolving a tractrix around its axis. Beltrami's paper convinced most mathematicians that the geometries of Euclid and of Lobachevsky are logically equivalent. In that work Beltrami used a differential equation corresponding to Gauss's equation. This has come to be known as Beltrami's equation, and later in this book we shall present the most recent developments in this area, solving Beltrami's equation at the critical point, where uniform ellipticity bounds are lost. This will necessitate the development of some considerable technical machinery to enable us to move away from the classical setting of uniformly elliptic PDEs to the case of degenerate elliptic equations. Beltrami's equation and its solutions, the quasiconformal mappings, have found a home in virtually all aspects of modern complex analysis, from the
- 24. 2 Introduction and overview theory of Riemann surfaces and Teichmiiller and Moduli spaces to more recent developments such as holomorphic dynamics and three-dimensional hyperbolic geometry. We hope the developments presented in this book encourage new applications of quasiconformal mappings in these areas. It has been nearly 200 years since Cauchy created the basic framework of complex function theory, principally during the years 1814-1831 [3061, since when the theory of conformal mappings and analytic functions has expanded in many different directions, far too numerous to relate here. This theory lies at the foundation of virtually all of modern analysis. Moreover, practical applications, such as in fluid flow, hydrodynamics and more modern areas of control theory, robotics and dynamical systems, abound. Geometric function theory (GFT) in higher dimensions is largely concerned with generalizations to R" of aspects of this theory of analytic functions of one complex variable, particularly the geometric and function-theoretic properties. We hope to give the reader a clear picture of these connections. In this sense GFT has been quite a successful theory, with many diverse applications. The category of maps that one usually considers in the higher- dimensional theory are, as in the planar case, the quasiregular mappings, or, if injective, quasiconformal mappings. Both kinds of mappings have the character- istic property of "bounded distortion" and solve PDEs closely analogous to the Cauchy-Riemann and Beltrami equations. Moreover, these mappings preserve the natural Sobolev spaces which arise in consideration of the function theory and PDEs on subdomains of R", or more generally n-manifolds. More recent developments have emphasized the connections between quasi- conformal mappings, harmonic analysis and PDEs. This connection is an import- ant aspect of our book. And that is why we depart from the usual theory of quasiconformal mappings quite early on and develop the theory of mappings with finite distortion. Again, the motivation here is to move into the realm of degen- erate elliptic equations where important applications lie. Usually, however, some control of the distortion functions (or equivalently the ellipticity bounds) will be necessary to achieve concrete results. These often take the form of integral estimates in some Lebesgue or Sobolev space. As mentioned, the governing equations for mappings of finite distortion are non-linear first-order systems of PDEs closely related to the Cauchy-Riemann equations and the complex Beltrami equation. There are also related second-order equations. For example, the components of an analytic function are harmonic, while those of a quasiregular mapping are "A-harmonic". In this way such well- known non-linear differential operators as the p-Laplacian and the associated non- linear potential theory arise naturally. There is also a close analogy between the analytic aspects of the theory of holomorphic functions and higher-dimensional theories of mappings of finite distortion. As we shall see, this analogy is particu- larly pronounced in even dimensions. A fruitful idea when studying quasiregular mappings, or more generally map- pings of finite distortion, is to view them as conformal with respect to certain
- 25. Introduction and overview 3 measurable metric or conformal structures. Indeed, it is from this point of view that the Beltrami equation initially arose and it is a view which we adopt in the last few chapters, where we present a selection of topics that represent fairly recent developments in a different direction from the analytic development in the first part of the book. Many of these notions, ideas and results extend to manifolds, and accordingly, while we do not develop this aspect in full, the reader should note that all the machinery we set up is ready for these developments. It is a deep result of Sullivan that all topological n-manifolds (n 54 4) admit quasiconformal structures, and we discuss this later in our overview. Thus one is able to do analysis on purely topological objects, relating topological and analytical invariants. For instance, Donaldson and Sullivan have developed a measurable Yang-Mills theory [78], there are analogues of the Atiyah-Singer index theory [321], and there is also the recent work of Connes, Sullivan and Teleman [73] developing the theory of characteristic classes in this setting. Quasiconformal mappings provide a class which lies between homeomorph- isms and diffeomorphisms. Mappings of finite distortion are even more flexible. Many constructions in analysis, geometry and topology rely on limiting processes. The compactness properties of families of mappings with finite distortion make them ideal tools for solving various problems in n-dimensional analysis and topology. For instance, in studying deformations of elastic bodies and the related extremals for variational integrals, mappings of finite distortion are often the natural candidates to consider because they are closed under uniform convergence, whereas the limit of a diffeomorphism need not be smooth nor even a homeomorphism. In this book, we present a considerable number of such compactness results for the class of mappings of finite distortion, see Chapter 8. In recent years there has been another well-known theory of mappings (referred to as deformations) whose ideas have come to the core of geometry and analysis. This is the non-linear elasticity theory of Antman, Ball and Ciarlet [12, 21, 22, 66], building on earlier work of Green [118, 119]. The theory was founded by the eighteenth-century mathematicians Bernoulli and Euler, who were concerned with the practical problems of mathematics and physics of that time. Nowadays the theory of elasticity studies mappings (in certain Sobolev classes) which minimize stored energy integrals. These mappings are not always quasiregular, but the governing PDEs are the same. It is necessary to study non- linear equations to observe certain physical phenomena such as bifurcation and phase transition. In particular, the Jacobian determinant (a highly non-linear geometric object itself) of these mappings has been subjected to a great deal of investigation. Its higher integrability properties were already recognized in the celebrated paper of Gehring in the 1960s, where he discovered the "reverse Holder inequalities". In this monograph we shall give a comprehensive account of higher integrability properties of Jacobians and other, more general, non-linear quant- ities which arise naturally in the L" theory of differential forms.
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- 27. We meet, for instance, with nations enjoying very ancient civilisations, like the Hindus and the Chinese; some, like the Mahomedans, under the power of a religion which they hold with the utmost tenacity of enthusiasm; others again, like the Buddhists, in proud possession of a philosophy and a literature that fully satisfies them. It is in such cases that the Gospel is confronted with its greatest difficulties. In conjunction with these conditions, others of a social character are sometimes found, that greatly increase the difficulties of the situation, as, for instance, where large communities are hedged round with the restraints of caste, which, while they secure them in the exclusive enjoyment of rank, influence and privilege, greatly cripple them in respect of liberty of conscience and conduct. To win people to the Gospel from such conditions has always been a difficult task, for it usually requires them to give up all that human beings ordinarily value most. But in the case of races like the Karens of Burma, the Pariahs and other low castes of India, and the negro slaves of the West Indies, Christianity finds human beings suffering from special disabilities, a lowly people, shut out, by the selfishness of those above them, from all the ordinary chances of bettering their lot, ill-used, oppressed, enslaved, kept in unlettered ignorance, deprived of all that makes life worth living. When the Gospel messenger speaks to them hopefully of a better state of things, and holds out a helping hand, it is evident, even to their dark minds, that this is their one chance of improvement, both in temporal and eternal things. They have everything to gain and almost nothing to lose by embracing the Gospel, and the consequence is that the success of the Gospel amongst such races is usually rapid. Another class of races there is, consisting of tribes wild and barbarous, beyond the confines of civilisation, and from time immemorial left to themselves, whose state of primitive savagery precludes the possibility of any elaborate form of religion, quite unlettered, and without a written language. Such are many of the races of the interior of Africa, many of the hill-tribes of Asia, and the inhabitants of the groups of islands in Polynesia. Here, again, are
- 28. found the conditions generally favourable for a rapid ingathering, notwithstanding their extreme barbarism and coarse brutality at first, amounting sometimes to cannibalism. For even the savage is conscious before long that he has something to gain by adopting the ways of civilisation. Where mission work has been conducted with perseverance in such countries it has always been successful. When we have fully recognised the mighty power of the Holy Spirit, to whose gracious influences we are indebted for all Gospel success, and when we have said all we have to say about different methods and men, the student of missions will still feel that he has not fully accounted for the marked diversity in the successes; he must also take account of the social and economic conditions of the different races, when the Gospel addresses them, and the hold their own religions generally have upon their minds. For the Gospel is like every other force in the universe, whether moral or physical, in this, that it always proceeds with most energy along the track of the least resistance; and he will find, if he carefully studies the matter, that the difficulties arising from social disabilities, and from a low state of civilisation, are not the greatest possible hindrances to the Gospel. In the approaching revival of missionary activity and enthusiasm these questions are sure to receive more careful attention; and when these problems come to be considered, Burma with its different races will contribute not a few interesting facts and experiences. The success of the Gospel amongst the Karens causes one to look wistfully at some others of the frontier mountain races of Burma. The religious views of all these primitive tribes are of much the same type, and their religious observances, what few they have, are similar. Their religion consists in the worship of nats or demons. They believe all nature is filled with nats; every stone, and tree, and pool, and breath of air has its spirit inhabiting it; and these nats are malevolent in their nature. Their religious observances consist not so much in worshipping them, as in propitiating them by means of offerings. They practise no regular system of worship, but consult the nats occasionally, whenever things do not go well with them, or
- 29. whenever there seems special reason to fly to the supernatural for guidance. Thus they have not much to cling to in the way of a religion, and their life and surroundings are so barbarous as to appear, even to themselves, obviously capable of improvement. In the north of Burma, on the mountains in the neighbourhood of Bhamo, are found the Kachins, a warlike hill people who have, since the annexation of Upper Burma, given the British some trouble by their raiding propensities. Amongst them the bones of sacrificed animals and other articles are placed outside the villages, to prevent the nats from entering in search of victims. It is believed that by this means their attention is called off. Some of the Kachins have taken to coming down from the hills and settling in Bhamo for work as labourers; and a successful work is being carried on by the American Baptist Mission there. Since the annexation a good deal has been done in the way of exploring the country, and bringing to light interesting facts with regard to these barbarous tribes on our frontiers. Lieutenant R. M. Rainey, Commandant of the Chin Frontier Levy, has published some interesting notes of his observations amongst the Chin tribes bordering on the Yaw country in the Pakokku district. The following facts are largely culled from his notes, many of them having been corroborated by what the writer and a missionary companion witnessed, in a recent visit to the tribe of the Chinbôk Chins, living nearest to the district described. The Chins of that region consist of various tribes all more or less distinct in language, and to some extent in customs, the Weloung Chins, the Boungshès, the Chinbôks, the Yindus, and the Chinbôns. No less than eight different dialects are spoken by these tribes, the Chinbôk language itself subdividing into three. There is no attempt at any system of laws or government amongst them, beyond the fact that they have something of a village system, and there are certain customs which all observe. Quarrels are wiped out with blood. Their religion, in common with that of all the other mountain tribes of the frontiers, consists in propitiating and
- 30. consulting the nats. For this an animal must be slaughtered—a buffalo, a bullock, a goat, a pig, a dog, or a fowl. The slaughtered animal is always afterwards eaten. In consulting the nats they observe the direction in which the blood of the sacrificed animal flows; this and similar omens are observed and acted upon. When raiding, or on a journey, or passing through a notoriously unhealthy jungle, sacrifices are frequently made, the animals being taken with them on purpose. Dogs are preferred for this object, as they follow, and require no carrying or leading. If the omens prove unfavourable they fear to carry out their purpose. Raids are frequently abandoned in this way at the last moment, and after they have travelled long distances. If, when the omens prove unfavourable, the parties are nevertheless desirous of accomplishing their purpose, as for instance in the case of an intended marriage, the nats are periodically consulted until they are favourable. This must always happen in time if they are only consulted frequently enough. The Chins are very much given to drunkenness, and are inclined to make of any and every incident a special occasion for getting drunk. A visitor, a birth, a marriage, a death, a case of sickness, are all possible and likely occasions for a carousal. In this worship of Bacchus they differ essentially from their Buddhist neighbours; but they may fairly claim to resemble in that respect many individuals of a distant race, and a race laying claim to a far higher civilisation. They have a novel mode of drinking the rice beer they manufacture for these occasions. The liquor is stored in jars standing two feet in height, and half full of fermenting grain. A hollow bamboo, the thickness of one’s little finger, is thrust into the jar and pressed well down into the grain. The company sit round and take sucks in turn. Of medicine and surgery they know nothing. When they fall sick they make no attempt at medicine, but merely consult the nats to ascertain the result, and propitiate them to avert the calamity. Scarcely any clothing is worn by the men, and that of the women, though sufficient for mere decency, is scanty, the legs being entirely
- 31. bare. They are all fond of ornaments. Necklaces of beads of all kinds are much worn, cocks’ feathers appear in the topknots of the men, and a kind of brass skewer is worn in the hair. They are also fond of wearing deer’s teeth and cowries. Telegraph wire, a new importation into their territory, forms a great temptation to them, inasmuch as a few inches of that metal, bent into a circle, forms a most becoming earring. Their weapons consist of bows and arrows, which they use with great dexterity. They often carry a short spear, and every man has a kind of weapon, which is dagger, knife and hatchet all in one, which sadly too often does murderous execution in their quarrels, and which, when not in use, is worn on the person in a bone scabbard consisting of the shoulder-blade of the buffalo. Their cultivation, though of a very rude description, is a laborious business. They have first to fell the jungle on the steep slopes of the hills, and after some months, during which it has had time to dry, they burn what has been felled. The grain is then sown without further preparation. They can only cultivate in the same place in this primitive fashion for two years together. In the third year the grass has grown so strong that cultivation is impossible. They then usually leave the land for five years, during which the jungle again grows up, when it is again cleared and cultivated as before. Their crops consist of rice and other grains, a considerable variety of yams and roots, including ginger, beans and vegetables, also cotton.
- 32. TATTOOING OF THE FACES OF CHIN WOMEN. The propensity of the Chins for raiding upon their weaker neighbours, and especially upon the Burman villages, is that which has compelled the British as the governing power to take account of them. Several military expeditions have had to be organised in order to punish this raiding, and to impress upon them the fact that it cannot be allowed. Many are the tales of the sudden descents of the Chins upon the peaceful villagers in the plains, robbing them of money, cattle and other property, and taking away prisoners, who are removed to the Chin villages, and held to ransom. If not quickly redeemed by their people they are often sold from village to village, which renders it difficult to trace and recover them. Many of these unfortunate captives have been rescued through our military expeditions. Perhaps the most extraordinary custom they have is that of tattooing the faces of their women. The process is commenced when they are young, and is gradually completed. Although the result is hideous to our eyes, it is said that the beauty of a woman is judged by the style in which the tattooing has been done. Thus fashion rules the world despite appearances and common sense. The Yindu women are tattooed in lines across the face. The Chinbôns tattoo jet black, and are the most repulsive in appearance, though often fair-
- 33. skinned. The Chinbôk method is to have several lines down the forehead, the nose and the chin; and the cheeks are covered with rows of little circles.
- 35. CHAPTER XI. BUDDHISM IN BURMA. The greater part of the inhabitants of Burma are Buddhists. The Burman race are so universally, except in the cases where Christianity has gained a few. It is in Burma that Buddhism is found with the least admixture of any other religion, and where it is followed with a more thoroughgoing devotion perhaps than anywhere else. Even the Burman, however, has never discarded in spirit, or even in form, the indigenous nat worship of his far-off ancestors. It may have little of outward appearance, but it remains side by side with Buddhism to the present day. In their numerous popular stories the nats play a prominent part, the wicked ones performing all manner of mischievous pranks, the good ones appearing at the opportune moment to succour the hero of the story, usually some “payaloung,” or incipient Buddha, for the moment in peril through the trials that have befallen him. This hankering after the nats is a significant fact. There is no God in Buddhism, and yet a man must have a deity or deities of some kind. The elaborate philosophy of Buddhism may occupy the intellect, and dominate the religious life, but it cannot satisfy this natural craving in man for God. Hence the worship and the fear of the nats, and the many superstitious ceremonies to propitiate them. And hence, too, if we mistake not, the strong tendency to plunge deeply into the occult, and to claim intimacy with the world of spirits, which characterises those Europeans and Americans who have discarded Christianity, and have devised for themselves a system fashioned on the basis of Buddhism, for their light and guidance. Buddhism has been well described as “A proud attempt to create a faith without a God, and to conceive a deliverance in which man
- 36. delivers himself.” Gautama, the future Buddha, and the founder of the Buddhist religion, was born at Kapilavastu, a town about one hundred miles from Benares, about 500 b.c. His father was the ruler of the Sakya tribe. Gautama early showed a disposition for a retired, studious, ascetic, contemplative life. His father wished to see him fit himself for the career of a prince, and heaped upon him every luxury, but in vain. At length we find the young prince, after many struggles between family affection and his view of duty, secretly by night leaving his home of luxury, his wife and child, exchanging his dress for the garments of a mendicant, and commencing his long quest after truth. Six years he spent in fastings and acts of penance. Then perceiving that mere ritual could bring him to no new conceptions of truth, he changed his method, and set himself to devise that system of philosophy which to this day is associated with his name. The ethics of Buddhism are grand, and for its noble conceptions of man’s duty it well deserves the title of the finest system of heathenism ever devised by man. But it fails altogether as a moral power. The account it gives of man’s nature, and the problem of life generally, though very elaborate, is erroneous and misleading. It knows nothing of a Divine Creator and Father, a Divine Saviour, or a Divine Regenerator. It proclaims no God, offers no Gospel of glad tidings, enjoins no prayer (in our sense of the word, as petition), sets forth no sacrifice for sin, holds out no hope of Divine help, no saving grace, no pardon, no renewal. Man must work out everything by his own endeavours. For forty-five years Buddha lived to preach his doctrines, winning many converts, and he died at over eighty years of age greatly revered.
- 37. “IMAGES OF BUDDHA ARE EXTENSIVELY USED.” That Buddhism is an uninspired system of teaching is most clearly indicated by its attempts at natural science. We need nothing more than a glance at these absurdities to dispose at once of Buddha’s claim to omniscience. His geography followed that of the Hindus, and was no improvement upon it. Its only virtue is that it is very liberal with numbers. It has its countless worlds, in the centre of which is the mountain called Maha Meru, 1,344,000 miles in length, the same in breadth, the same in depth beneath the sea, and rising to the same height out of it. Its teaching upon such matters as eclipses, earthquakes and the like, consists of the wildest of guesses. It may be well to give the reader a brief outline of the religious teachings of Buddhism. Buddhism denies the creation of the world. Matter is eternal, and all the changes attending it are caused and regulated by certain laws co-eternal with it. Matter and its laws are
- 38. not under the control of any being. Hence creation and a creator are out of the question. With such a formidable list of negations to begin with, it becomes a matter of no small interest to inquire out of what materials this vast system could possibly have been constructed. First, then, we have the Buddhist ten commandments. Five of these are binding upon all:— 1. Not to take life. 2. Not to steal. 3. Not to commit adultery. 4. Not to lie. 5. Not to take that which intoxicates. The other five are applicable only to the monastic order:— 6. Not to eat after midday. 7. Not to attend theatrical amusements, or dance, sing, or play on a musical instrument. 8. Not to use garlands, scents, or cosmetics. 9. Not to stand, sit, or sleep on a platform or elevated place. 10. Not to receive gold or silver. Besides these precepts there are many minor regulations, some of them entering very minutely into the life of the laity, and others the monks. There are rules for the conduct of parents and children, pupils and teachers, husband and wife, friends and companions, masters and servants, laymen and the religious order; in fact, considering the light Gautama possessed, the moral teaching of Buddhism is of a very high order. But what about the means of attaining to moral excellence? Here Buddhism, it must be confessed, is found wanting. To conceive of a high state of moral excellence is manifestly better within the reach of
- 39. man’s unaided mind, than to find out a way for the bulk of mankind in their frailty and sinfulness to reach it. In order to place before the reader any intelligible view of the Buddhist way of salvation, it is essential that we consider first its teaching concerning the nature and circumstances of man. Buddhism is thoroughly pessimistic in its outlook. It teaches that life is a misery, existence an evil. This doctrine is taught in the sacred books with a wealth and ingenuity of illustration worthy of a more gay and festive theme. The sentient being is “like a worm in the midst of a nest of ants; like a lizard in the hollow of a bamboo that is burning at both ends; like a living carcass, bereft of hands and feet, and thrown upon the sand.” All beings are “entangled in a web of passions; tossed upon the raging billows of a sea of ever- renewing existences; whirling in a vortex of endless miseries; tormented incessantly by the stings of concupiscence; sunk in a dark abyss of ignorance; the wretched victims of an illusory, unsubstantial and unreal world.” It is true these views of life do not seem unduly to distress the followers of Gautama. The Burmans, the best of Buddhists, are as merry and laughing a people as are to be found anywhere, and the burden of life rests not more lightly upon any people than upon them. Nevertheless such is the teaching. “Anaiksa, Doakka, Anatta” is the formula in Burmese: “Transient, Sorrowful, Unreal.” The monk muses on this in his monastery. The pious Buddhist repeats it to himself as he spends his spare time smoking and meditating on the bench at his door, or strolling idly about, telling off the beads of his rosary the while. Seeing that life is necessarily a misery, and existence an evil, the problem of life would seem to be how to bring existence to an end. The Christian would say wait for the release of death, but two formidable difficulties stand in the way, to prevent death proving any release—namely, Transmigration and Karma (Burmese Kan).
- 40. Transmigration constantly renews sentient existence in a countless succession of births and lives. Hence the polite form of the announcement of a death is that the deceased has “changed his state of existence,” that is, put off one existence and taken on another. This is not merely a polite form of speech, but more correctly embodies the popular belief than the mere statement that he has “died.” Moreover, in future births man may rise and fall in the scale of existences; and as human life and animal life are considered to be of the same nature, no difficulty is experienced in readily believing that a man may become an animal, or an animal may become a man in future births. Hence the scruple against taking any kind of animal life amongst the Burmans, extending even to vermin. Supposing transmigration to be true, it follows that if one kills any animal, large or small, even the smallest insect, he may be taking the life of his deceased grandfather, who has thus reappeared in the body. This universal belief of the Buddhists in transmigration was curiously illustrated quite recently in a court of justice in Burma. A mother and her son came one day to the magistrate of their district and expressed a desire to institute a suit. The case for the son, who was the plaintiff, was as follows. Some years before, a certain man, it was stated, had left in charge of the defendant some jewellery and a silk cloth for safe keeping. While engaged in repairing the roof of a house he fell off and died of the injury. The jewellery and cloth remained in the hands of the defendant, and the suit was now instituted to recover the same. What was the ground for this claim? Not that this boy or his mother were related to the deceased, but that the boy was that identical man in another birth. But how could he prove it? There was no difficulty in proving this, at least to the satisfaction of the Buddhists. The boy displayed upon his body certain marks, which those who knew the deceased said were precisely similar to marks he bore. The mother, by a comparison of, dates, sought to prove the date of the birth of the boy was just when it would be supposing his claim to be true. But the most convincing testimony of all was that
- 41. the boy distinctly remembered the whole of the circumstances happening in his former existence! The defendant admitted receiving the silk cloth, but denied all knowledge of the jewellery. He admitted that he believed the boy was the very man who left the cloth with him, and was willing to return it if the boy paid a small debt of eight annas borrowed on it by the owner. The boy said he remembered the eight annas, but also insisted on the jewellery. Unfortunately for him his good memory did not avail him; it was a British court of justice, not a Burmese, and the magistrate had to dismiss the case as extending to matters beyond his jurisdiction. Karma or Kan (Burmese), or Fate, as it is sometimes rather inadequately rendered, is that self-originating, self-operating, inflexible law which necessitates and causes the working out of the cumulative influences of merit and demerit; these separately producing in succeeding births their full and appropriate effects, extending through cycles of ages, the Kan being modified from time to time by the passage through these different births. Thus Kan is not in any sense a Divine Providence. It is a blind impersonal force that attends our destiny through all the course of our many existences, and makes us to reap in other births what we sow in this. It may be compared to a balance. In the one side we are always putting in acts of merit, and in the other side acts of demerit, and the Kan goes on determining which preponderates, and blindly producing its appropriate consequences until each has worked itself out to the pleasant or the bitter end. Undoubtedly this doctrine is a bold expedient for explaining the apparent anomalies and wrongs in the distribution of happiness and misery in this life; and although it is incapable alike of proof and of disproof, it fully satisfies those who can believe it. A child, for instance, is blind,—this is owing to his eye-vanity, lust of the eye in a former birth,—but he has also unusual powers of hearing; this is because he loved in a former birth to listen to the preaching of the law. Thus the theory can always be made to fit the facts, for it is derived from them. But it satisfies the Oriental mind none the less
- 42. for that, and it is the belief of millions of Hindus and Buddhists to- day. Nirvana (Burmese Neibbân) is the state of complete deliverance from further births and deaths. So long as existence lasts evil and suffering must continue, and there is no hope of blessedness until conscious individuality has become wholly eliminated, and the individual has arrived at that state where further births are no longer possible. This means practically annihilation; but it is so much easier to do wrong than to do right, and it takes so long for Kan to work out its result, that Neibbân becomes, by the ordinary way, so distant and so difficult of attainment as to be out of reach to the vast majority of the human race. If Buddhism ended there, and if nothing had been devised to relieve this strain of seeking after an all but hopeless and well-nigh impossible good, it would have been of all creeds the most pessimistic and miserable. The mind must needs have revolted from an outlook so gloomy, and we may safely affirm that it would in that case never have numbered its votaries by hundreds of millions as it does to-day. For it just amounts to this, that “Sin and its consequences follow man as the wheels of the cart follow the legs of the bullocks,” and there is no Saviour and no salvation that he can seek outside of himself. But just at this point the doctrine of works of merit steps in and offers its hopes to the Buddhist, and seems to bring the attainment of future good at once within the sphere of the practicable. According to this, man can be continually improving his Kan by so- called works of merit, and he may hope, with comparatively little trouble, to make his merits outweigh his demerits, and thereby improve his lot in future existences. See that row of waterpots under the shade of that great tree upon a dusty road, set upon a neat stand, with a neatly carved roof constructed over them, with a ladle to drink out of, and each of the pots covered with a tin cover to keep out the dust and insects. It is privately constructed and presented for public use, a work of merit;
- 43. all done to get what they are often thinking and talking about— koothoh. What is the meaning of all this lavishing on the monks of food daily, and various offerings, including almost everything except money, which they are under vows not to touch? Answer, koothoh. So with all alms and offerings to monks, to the poor, to dogs, or crows; so with good works of every imaginable description. You may acquire merit by conforming to the ceremonies, by attending the festivals, by listening to the reading of the Law, by striking the pagoda bells, by buying and lighting pagoda tapers, by plastering gold leaf on the pagoda, by contributing to the repairs of the sacred edifices, by showing lights at the festival of lighting in October, and by many, many ways. As might be expected, when the acquiring of merit is so important a matter, there are many avenues opened to it. Though of course you have not kept all the laws, yet if you have gone out of your way a little to do something more than keep one of them it gives you merit. The care for animal life offers great scope in that direction. An English soldier whilst fishing caught a tortoise and was taking it home, when a Burman met him, bought the tortoise for a rupee, and took it back to its native element. He would expect to gain merit by that. Men have been known to make a regular trade of snaring little birds in the jungle, and bringing them to the bazaar to sell to the merit seekers, who buy them merely to set them free. Many works of merit involve great expense, such as the digging of a well, the erection of a bridge, a zayat or rest-house, a monastery, a pagoda. Judging by the enormous number of these sacred buildings in Upper Burma, it would appear that this is a favourite way of seeking merit. The builder of a pagoda is honoured with a special title attached to his name, and he is understood to be in a fair way for Nirvana. This seeking after merit is practically the most predominant aim in Burmese religious life. So fixed is this belief in merit, that when the Burmans see the English so intent upon opening up the country, making roads and railways, metalling streets and lighting them, building hospitals and
- 44. markets, constructing irrigation works, and carrying out a multitude of other necessary and useful efforts of public utility, they measure us by their own bushel, and remark that there will be great merit to the Government and its officers by means of these things. What other motive could men have for taking so much pains and trouble for the public good, if not to accumulate merit? In elaborating this law relating to merit, Gautama was preparing the sheet anchor of his system. It is that mainly by which it abides, and retains its influence over its millions of followers until this day. “IN THE MORNING THE MONKS INVARIABLY GO FORTH CARRYING THE ALMS- BOWLS TO COLLECT THEIR DAILY FOOD FROM THE PEOPLE.” Every false religion, however, whatever master mind designed it, must show, somewhere or other, its weak places. It is manifestly a weak place in Buddhism that alms and works of merit may so easily outweigh whatever demerit may attach through real crimes and sins, and that, too, without any repentance or reformation on the part of the offender. This also makes the attainment of merit largely
- 45. dependent on the pecuniary means and influence at the disposal of the individual. A work may be very easy for a king or a rich man which would be utterly impossible for a poor man. To the Christian mind this seems very unequal and unfair, but to the Burman it presents no stumbling-block. Supposing we do see great inequalities in money, or any other temporal advantages that men possess. Be it so. It arises from differences in their Kan, and that depends on what took place in previous births. One’s Kan is not a thing to rail against, but to submit to. It might be thought that as Christianity is so evidently superior to Buddhism as a religious system, it should be an easy matter to get them to discard their religion and accept the religion of Christ. But this is very far from being the case. The superiority is not apparent to a mind sophisticated by a lifelong familiarity with only the one religion, and it is only, as a rule, perceived after a prolonged and impartial study and comparison of the two has opened the mind. This is the great reason for educational work. It is a very difficult matter to make the votaries of an elaborate system like Buddhism see the superiority of Christ over Buddha; they are more than contented with what they have. Besides this, we ought to remember that Buddhism has everything on its side that tends to make a religion powerful and influential. It has a concrete existence, and very much of outward and visible form and appearance; it is in possession; it has numbers, a voluminous literature, a definite and consistent system of philosophy. It has plenty of popular observances and popular enthusiasm. It is cleverly adapted to man’s natural desire to work out his own salvation. It is most powerfully sustained and buttressed in the regard and confidence of the people by its very numerous monastic institutions, which are recruited from all classes of the people, from the prince to the peasant, for every male Burman must be a monk, for a longer or a shorter time, at some period of his life.
- 47. CHAPTER XII. BURMESE RELIGIOUS INSTITUTIONS AND USAGES. The Burmans, like most nations of the East, are essentially a religious people, and pay great regard to the religious usages and institutions in which they have been brought up. Chief amongst these is the monastic institution of Buddhism. Buddha was not only a great philosopher and thinker, but a great organiser too, and he provided in the monastic system a social bond of union that knits the entire community together in the Buddhist faith. This is made more obvious by the fact that every male Burman must be a monk at some time in his life, for a longer or shorter period, otherwise the demerit attaching to him would so overbalance his merits, as to render it impossible for him ever to make any improvement in his future existences. His ill deeds would swell the sum of his demerits, but no act of charity or pious devotion would be recorded to his advantage. Hence, in Upper Burma, almost every youth dons the yellow robe and becomes a monk. It may be for a week, a month, or a season or two, or it may be for many years, or it may prove to be lifelong. The longer they stay in the monastery the more sanctity attaches to them. But the Buddhist monk, unlike some other monks, is at liberty to terminate his monastic vows at pleasure, and return to ordinary life. The monks reckon the continuance of their monastic condition by the number of Wahs spent in the monastery, the Wah being the annual recurrence of a kind of Buddhist Lent, extending from July to October. This recruiting of the monks from the entire population—so different from Hinduism, which acknowledges a rigidly exclusive, priestly caste—immensely strengthens the hold Buddhism has on the people, and widens the popular basis upon which it rests. In my
- 48. missionary life amongst the Hindus in Ceylon, I have observed in reading and expounding the parable of “The Good Samaritan” to a heathen congregation, a great readiness to apply, of their own accord, the cases of the Priest and the Levite, who passed by on the other side, to their own Brahmin priests, and they were always ready to take sides against them as quite a separate caste; but there can never be the same alienation between monks and laity in a Buddhist land like Burma, where the monks are their own kith and kin. The monasteries are very extensively spread over the country. Mandalay, at the time of the annexation, was officially stated to have close upon 6,000 monks, and you can visit scarcely any town or village, however small or remote, which has not its monastic establishment. The monastery is always the best building in the place, and has the cleanest enclosure of any house in the village, and there is an air of sanctity and repose about it. The monks are very approachable. The stranger, whether native or foreigner, is always made welcome; indeed, that is a characteristic of the Burmans everywhere, that they receive strangers freely and affably, and being free from those caste scruples so usual amongst the Hindus, one is not for ever fearful of transgressing their notions of propriety, or unwittingly hurting their dignity. As the monasteries are spacious, and often supplied with additional zayats or rest-houses, it will rarely happen in travelling that they will be unable or unwilling to assign the stranger some humble place of rest, where he may tie up his pony, eat his food, and spread his mat and pillow for the night. To the poor and destitute the monastery is a place of relief, where they can always hope to obtain a little food out of that which is daily given to the monks in their house to house morning visits. It must be frankly admitted that the monasteries of the country do a useful work in the way of imparting elementary education. To them is chiefly due the creditable fact that there are comparatively few of the men who cannot read and write; and this does much to bind the people to the support of the Order. But the education scarcely ever goes beyond the most elementary stage. They learn to read Burmese, and they learn to repeat a few Pali prayers and forms of
- 49. devotion. Pali is the sacred language; very few even of the monks understand the meaning. On the other hand, the monks’ life is a very idle one. They live in perfect ease, all their wants are supplied by the people, and they are not expected to work at all, except some of them at teaching. There are usually far more of them in the monastery than are required for that purpose, so that they spend a vast amount of idle time, and it is thought by many that the indolent, easy-going habits, and the lack of discipline and enterprise the Burmans display as a nation, is largely owing to the idle life of the monastery, which is continually before their eyes, for there they receive their teaching when young. The Buddhist monk is not a minister of religion in our sense. He has no pastoral charge. He is for himself, and for his own deliverance, and the merit he acquires he shares with nobody. He may occasionally be called to attend this or that function, when the presence of a monk is customary, or he may expound the law occasionally, if he so choose, reciting some of the sacred writings for that purpose; but he undertakes no responsibility for the guidance of the souls of others. In Buddhism a man must save himself, and nothing that a monk or any one else may do can alter the balance of his merits and demerits. Even if the monk be summoned to the couch of a dying man, as he is sometimes, it is not that he may speak words of consolation, or offer him the comforts of religion. It is merely that the presence of the holy man may drive away the evil spirits that would be liable to haunt the place on such an occasion. The habits of the monastic Order are very simple. In the morning, after the few Pali prayers have been uttered, the monks invariably go forth through the village, attended by the boys carrying the alms- bowls, to collect their daily food from the people. Not that they beg. There is no occasion for that. Their rules forbid them to ask; and in going from door to door amongst their own people they do not ask. But privately, I must own, I have found occasionally amongst the Buddhist monks of my acquaintance some of the most arrant cadgers I ever met with. Few, indeed, are the matrons who do not
- 50. put something in the way of food in the alms-bowl. Nor do they thank the people for what they receive. They would never think of doing so. In fact, the obligation is all on the other side. The monks are conferring a favour by giving the people the opportunity to do this work of high merit by means of their alms. A useful hint, by the way, to collectors for good and useful objects in England! In their walks abroad, and in the performance of such functions as bring them into mixed companies, many of the monks carry a large palm-leaf fan in their hands, in order that, as celibate ascetics, they may shut off the sight of feminine charms from their eyes. The education given at the monasteries is very poor, but the acquisition of any learning at all by the children is a matter for wonder, when we consider how poor the instruction is. What they do succeed in learning is not so much by means of teaching, as we understand it, but is almost entirely due to the system of noisy repetition of the lessons, at the full pitch of their voices in unison, in which all the children engage, the elder ones leading, and the younger following. For this reason the little learning imparted at these schools is of a mechanical sort, and lacks intelligence. Arithmetic is very low indeed. Geography, if taught at all, must of course square with the orthodox Buddhist cosmogony; and as there is much that is doubtful about that, it is perhaps best left alone, and is accordingly. Burmese history is abundant in quantity, but in quality it only consists of what we call fiction, and has but a poor foothold upon fact, and is left out of the curriculum. All other branches of study are unknown, except a little of Pali in the form of devotions, which, however, is mostly taught in mere parrot fashion.
- 51. “THERE THE PEOPLE ASSEMBLE OF AN EVENING, AND ARE TO BE SEEN IN THE OPEN SPACE AROUND THE PAGODA, ON THEIR KNEES IN THE OPEN AIR, REPEATING THEIR DEVOTIONS IN PALI.” The Director of Public Instruction in Burma told me a good story of his first visit to Mandalay. He had been calling on the great Thathanabine or Buddhist Archbishop of Burma, and had sought to impress upon that venerable ecclesiastic the desirability of improving the education given at the monastery schools. He mentioned arithmetic and geography as very desirable subjects to be taught, offering to supply teachers already trained and able to teach them. One of the attendant monks, an elderly brother of the yellow robe, remarked that for his part he could not see any great need for learning geography, especially now that the English Government had been good enough to construct a railway. “If you want to go anywhere all you have to do is to take your ticket and get into the train.” Where was the use of learning geography?
- 52. The honour paid to the monks by the people is quite extraordinary. In the Burmese language the commonest acts of life as performed by the monks are spoken of with respectful expressions, which are never applied to similar acts as done by the common people. The oldest layman honours the youngest monk, and gives place to him. The ordinary posture before a monk is down on their knees, and often on their elbows also, with the palms of the hands joined together, and raised as if in supplication, and the title “Paya” is used—the very name which has to do duty for the deity. An instance is on record of a venerable monk being called from Mandalay to settle a dispute between two parties concerning some religious point, in a town on the banks of the Irrawaddy. On his arrival the whole population lined both sides of the path up to the monastery, and kneeling, they loosed down their long black hair, for the men as well as the women wear it long, and spread it across the path, so that he walked all the distance from the river bank to the monastery on human tresses. The pagodas are the ordinary resorts of the people as places of worship; not all of them, however, for the great majority are merely erected as works of merit, and never attain any celebrity as places of worship; only the chief and most notable shrines. There the people assemble of an evening, and are to be seen in the flagged open space around the pagoda, on their knees in the open air, repeating their devotions in Pali. Though many of them come together, it is not of the nature of congregational worship, nor is any one appointed to lead their devotions. It is each one for himself. There is no prayer in our sense of that term, that is, petition. With no God to address, what place is there for prayer? Buddhism knows no higher being than the Buddha, and he is gone, twenty-four centuries ago, into Nirvana. The sentences they utter in Pali consist of expressions in praise of Buddha, the Law, and the Monastic Order. Images of Buddha are extensively used, but for all that, the people can hardly be called idolaters. The burning of candles and of incense at worship time is customary.
- 53. The Burmese “duty days,” of which there are four in the month, are observed on the eighth of the crescent, the full, the eighth of the waning, and the change of the moon. These are kept more strictly as worship days during what is called the Wah than at any other time. That is the period from July to October, which is observed as a time of special fasting and solemnity, ever since the days of their founder, who used to spend this, the rainy season, when travelling about in India is scarcely practicable, in retirement and meditation. During the Wah there is a cessation of all festivities, and of the theatrical performances of which the Burmans are so fond. At the end of the Wah there is a time of general rejoicing. For some days before amusements are observed to be in progress in the streets. Effigies of animals, very well executed, are carried about. Here a buffalo of gigantic size, made of some light material, cunningly finished and coloured to the life, with horns and hide and all complete, is seen walking about on two pairs of human legs, the said legs being clad in the very baggy dark trousers worn by the Shans; its head balanced so as to swing with the walk in a most realistic and natural manner. Yonder, in the Chinese temple, a huge pasteboard demon is seen disporting himself, with head of frightful aspect and enormous size, and body of cloth. You may freely walk in; and as you look around and admire the excellence of the building and the expensive and choice furniture, lamps and decorations, you may also see the huge creature writhing about, with all manner of contortions, to the deafening din of drums and the clash of cymbals. Somehow Orientals seem to be able to combine amusement with devotion. My three little children who have walked in with me, scared almost out of their wits with the noise, and still more with the portentous sight of the demon, promptly take to their heels and rush out of the temple, and cannot be induced to return, so I go out in search of them. At the corner of the next street an enormous representation of a tiger, ten times life size, in teeth and claws complete, and with a most ferocious aspect, has been glaring at the passers-by for some days. And, as you look, here comes a rude likeness of a gigantic lady
- 54. ten feet high, who, however, seems to move along very ungracefully, and bows very stiffly in acknowledgment of the cheers of the crowd. The Chinese are particularly fond of getting up a very brilliantly executed figure of a serpent, in great splendour and in very bright colours, many yards long, which is borne high overhead through the streets on these occasions, with quite a procession. This particular show seems to afford scope for high art in representing the wrigglings of the monster as it is carried along. But this is all only preparatory to the festival called Wah-gyoot (literally “the release from the Wah”). It is a festival of lights. For three nights the whole city of Mandalay is one blaze of illumination. Every house has its complement of candles or oil lamps; the rich in keeping with their means, and the poor according to their poverty. At that season the air is still, there is little or no wind, all the lights are out of doors and burn brightly. The streets are lit up with candles at every ten paces; the pagodas are effectively illuminated with hundreds of lights far up into their spires. Little children are trundling extemporised carts with bamboo wheels, each carrying a tiny illumination, covered with a lamp of thin, coloured paper. In addition to the house illuminations, paper lanterns are quite the fashion in China Street, where the well-known ingenuity of John Chinaman produces fantastic shapes in various colours, representing sundry animals, fishes, ships and what not. On the great river, as soon as it grows dark, the villagers row out into the middle of the stream and set adrift multitudes of oil lamps, each fastened to a little float of bamboo or plantain stem. Thousands of them are sent out by each village, so that the whole Irrawaddy is one blaze of twinkling lights. Another very prominent and popular festival of the Burmans is the Water Feast, which occurs at their New Year in April. For two or three days at that time “the compliments of the season” consist in walking up to you in the street, or even in your own house, and discharging a jar of clean water over you, with the expression, “I will do homage to you with water”; and it would be considered very bad form to show any resentment for this kind and polite attention. It is
- 55. obvious that such a custom as this must afford great scope to the rollicking Burmans of both sexes. It leads to abundance of larking and merriment in the streets. Everybody who ventures forth stands a great chance of a thorough drenching. Fortunately it occurs in April, the time of the sun’s greatest power, and the sweltering heat renders it less of an inconvenience than it would be in a colder climate. There is nothing the Burmans are more scrupulous about than the taking of life. A mother has been seen to pick up the scorpion that stung her child, between two pieces of bamboo, and merely drop it gently outside the door. Twice when I have found a deadly cobra lurking about the house where the children were playing—the most venomous of snakes, whose bite is death—and have asked a Burmese servant to help me to kill it, he has declined, and I have had to kill it myself. But though the Burman will not kill a snake, he will not scruple to take it home to cook and eat it after some other person has killed it. Animal food seldom comes amiss to them, whether it has been killed by another or has died of itself. They are not very choice in their food. Mandalay swarms with thousands of half-starved, mangy, miserable animals—nobody’s dogs. No matter how they increase and multiply, no Burman is willing to “put them out of their misery”; the firm belief in transmigration prevents this. I have known half a dozen such dreadful creatures quarter themselves uninvited on the Mission premises. One of the half-dozen, a savage brute, living under the school on the Mission premises, one day bit a little Burman boy, and tore his bare arm very badly. This was too much for me. Fearing it might do further mischief, and might even be mad, I waylaid and shot it. The Burmans thought I had done very wrong. Their tender care for animals often appears in touching forms. I have noticed a Burman coolie engaged in mixing mortar, on finding he had brought a number of tadpoles from the neighbouring pond in his bucket of water, take them all out with great care, and carry them back to the pond, though it was 150 yards away and he had to go on purpose. And yet, so strangely inconsistent is human nature, there are
- 56. perhaps few countries in the world, with any pretensions to civilisation, where human life is held so cheap as in Burma, and where the people have commonly such a propensity to the crime of dacoity or robbery with violence, and often with murder. And yet, again, with strange inconsistency, the coarse and hardened criminal, the Burman dacoit, who has imbrued his hands in his neighbour’s blood more than once, will scruple to harm the vermin that infests his couch. Some of the great Buddhist shrines in Burma are buildings of wonderful magnificence. The Shwê Dagohn Pagoda at Rangoon is one of the most important and sacred. It is considered to be over two thousand years old. Originally it was very small, but now it rises to a height of 370 feet, or a little higher than St. Paul’s Cathedral, and is a quarter of a mile in circumference at the base. It is situated on the top of a very high hill, of which the summit has been, at vast labour and expense, made into a level platform, and carefully paved. This immense platform is partly occupied by many smaller pagodas, resting places for worshippers, and chapels containing colossal images of Buddha; and considerable open space is left for the immense crowds of worshippers that assemble there. In the centre rises the great pagoda in the usual bell shape, one vast, solid mass of masonry terminating in a spire. Four flights of stone steps lead up from the plain beneath, one on each side of the hill. On the summit of the pagoda is the htee, or gilt iron framework in the form of an umbrella, with multitudes of gold and silver bells, richly bejewelled, which tinkle with every passing breeze. The htee was presented by King Mindohn, the father of King Theebaw, and cost £50,000. The pagoda itself with the adjacent buildings must have cost, from first to last, a fabulous sum. This pagoda, like many others of the principal ones, is covered with pure gold leaf. Every few years it has to be regilt. Sometimes this has been done by some particular king, as a great work of merit. One king is said to have spent his own weight of gold upon it. In 1887 there was a regilding by public subscription. The accounts when published showed an expenditure of some £9,000; and this money, be it known to all Christians, was
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