DIFFERENTIAL GEOMETRY

http://dx.doi.org/10.1090/pspum/027.2

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

VOLUME XXVII, PART 2

DIFFERENTIAL GEOMETRY

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1975

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT STANFORD UNIVERSITY STANFORD, CALIFORNIA

JULY 30-AUGUST 17, 1973

EDITED BY

S. S. CHERN and R. OSSERMAN

Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-37243

Library of Congress Cataloging in Publication Data

Symposium in Pure Mathematics, Stanford University, 1973. Differential geometry.

(Proceedings of symposia in pure mathematics; v. 27, pt. 1-2)

"Final versions of talks given at the AMS Summer Research Institute on Differential Geometry."

Includes bibliographies and indexes. 1. Geometry, Differential—Congresses.

I. Chern, Shiing-Shen, 1911- II. Osserman, Robert. III. American Mathematical Society. IV. Series. QA641.S88 1973 516'.36 75-6593 ISBN 0-8218-0248-8 (v. 2)

Copyright © 1975 by the American Mathematical Society

Printed in the United States of America

OE

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission

of the publishers.

CONTENTS

Preface ix

Complex Differential Geometry

Riemann-Roch theorem for singular varieties 3 By PAUL BAUM

A construction of nonhomogeneous Einstein metrics 17 By E. CALABI

Generalizations of the Schwarz-Ahlfors lemma to quasiconformal harmonic mappings 25

By S. I. GOLDBERG AND T. ISHIHARA Holomorphic mappings to Grassmannians of lines 27

By MARK L. GREEN Some function-theoretic properties of noncompact Kahler manifolds 33

By R. E. GREENE AND H. W U Differential geometry and complex analysis* 43

By PHILLIP A. GRIFFITHS On the curvature of rational surfaces 65

By NIGEL HITCHIN Holomorphic extension for nongeneric CK-submanifolds 81

By L. R. HUNT AND R. O. WELLS, JR. Holomorphic extension theorems 89

By PETER KIERNAN Residues and Chern classes 91

By JAMES R. KING Some classical theorems for holomorphic mappings into hyperbolic

manifolds 99 By MYUNG H. KWACK

*General lecture given at the Institute.

v

VI CONTENTS

Some results in/-structures motivated by the Cousin problem 105 By RICHARD S. MILLMAN

Holomorphic equivalence and normal forms of hypersurfaces 109 By JtJRGEN MOSER

On compact Kahler manifolds with positive holomorphic bisectional curvature 113

By TAKUSHIRO OCHIAI

Partial Differential Equations

On the size of a stable minimal surface in R* 127 By J. L. BARBOSA AND M. DO CARMO

Geometry of the spectrum. I* 129 By M. BERGER

Constant scalar curvature metrics for complex manifolds 153 By MELVYN S. BERGER

Sobolev inequalities for Riemannian bundles 171 By M. CANTOR

Eigenfunctions and eigenvalues of Laplacian 185 By SHIU-YUEN CHENG

Minakshisundaram's coefficients on Kaehler manifolds 195 By HAROLD DONNELLY

The spectrum of positive elliptic operators and periodic geodesies 205 By J. J. DUISTERMAAT AND V. W. GUILLEMIN

Random walk on the fundamental group 211 By JAMES EELLS

Linearization stability of nonlinear partial differential equations 219 By ARTHUR E. FISCHER AND JERROLD E. MARSDEN

The spectral geometry of real and complex manifolds 265 By PETER B. GILKEY

Spectral geometry and manifolds of constant holomorphic sectional curvature 281

By PETER B. GILKEY AND JONATHAN SACKS Whitney's imbedding theorem by solutions of elliptic equations and

geometric consequences 287 By R. E. GREENE AND H. W U

Fourier integral operators from the Radon transform point of view 297 By V. GUILLEMIN AND D. SCHAEFFER

A hierarchy of nonsolvability examples 301 By C. DENSON HILL

Extending isometric embeddings 307 By H. JACOBOWITZ

•General lecture given at the Institue.

CONTENTS Vll

Prescribing curvatures 309 By JERRY L. KAZDAN AND F. W. WARNER

On symplectic relations in partial differential equations 321 By BOHDAN LAWRUK

On periods of solutions of a certain nonlinear differential equation and the Riemannian manifold 0\ 327

By TOMINOSUKE OTSUKI Singularities and the obstacle problem 339

By DAVID G. SCHAEFFER Harmonic mappings of sphere 341

By R. T. SMITH Holomorphic jR-torsion for Lie groups 343

By NANCY K. STANTON

Homogeneous Spaces

The first eigenvalue of the Laplacian on manifolds of nonnegative curvature 351

By I. CHAVEL AND E. FELDMAN The generalized geodesic flow 355

By LEON W. GREEN The eigenfunctions of the Laplacian on a two-point homogeneous space:

integral representations and irreducibility 357 By SIGURDUR HELGASON

The cohomology ring of SO(2n + 2)/SO(2) x SO(2n) and some geometrical applications 361

By HON-FEI LAI Representations of linear functional on HP spaces over bounded

homogeneous domains in CN (N > 1) 363 By JOSEPHINE MITCHELL

Noncompact riemannian manifolds admitting a transitive group of conformorphisms 373

By MORIO OBATA Critical sets of isometries 375

By V. OZOLS Partial spin structures and induced representations of Lie groups 379

By JOSEPH A. WOLF

Relativity

Einstein-Maxwell theory and the structure equations 385 By GEORGE DEBNEY

Lorentzian manifolds of nonpositive curvature 395 By F. J. FLAHERTY

Vlll CONTENTS

General relativity* 401 By ROBERT GEROCH

Gravitational waves and averaged Lagrangians 415 By A. H. TAUB

Gravitational collapse 425 By John A. THORPE

Indexes

Author Index 433 Subject Index 439

•General lecture given at the Institute.

Preface

The papers in these PROCEEDINGS represent the final versions of talks given at the AMS Summer Research Institute on Differential Geometry, which took place at Stanford University, Stanford, California, from July 30 to August 17, 1973. This Institute was made possible by a grant from the National Science Foundation. The organizing committee consisted of Raoul H. Bott, Eugenio Calabi, S. S. Chern, Leon W. Green, Shoshichi Kobayashi, Tilla K. Milnor, Barrett O'Neill, Robert Osserman, James Simons, I. M. Singer, with the coeditors serving as cochairmen.

The activities were divided between general lectures and seminar talks. In these PROCEEDINGS the general lectures have been distributed among the various sem­inars, according to their subject matter. Each part of the PROCEEDINGS consists of a group of seminars, whose titles and chairmen are as follows:

Part 1: Riemannian geometry (J. Cheeger) Submanifolds (K. Nomizu) Foliations (B. L. Reinhart) Algebraic and piecewise-linear topology (T. F. Banchoff and H. R. Gluck) Miscellaneous (B. O'Neill and J. Simons)

Part 2: Complex differential geometry (S. Kobayashi) Partial differential equations (J. L. Kazdan and F. W. Warner) Homogeneous spaces (J. Wolf) Relativity (T. Frankel)

Generally papers are included in the seminars in which they were presented, although in certain cases the contents would make them more appropriate in another section. In cases where a complete version of the talk appears elsewhere, only an abstract is included here, together with a reference to the full paper.

IX

X PREFACE

A list of open problems submitted by participants was compiled by Leon Green. These are included at the end of Part 1 in the Miscellaneous Section.

We should like to think the seminar chairmen, and also the secretarial staff: Dorothy Smith and Muriel Toupin of AMS as well as Catherine Lowe and Eli­zabeth Plowman of the Stanford Mathematics Department, all of whose tireless efforts were a large factor in the success of the Institute.

S. S. CHERN ROBERT OSSERMAN

JANUARY 1975

INDEXES

AUTHOR INDEX

Roman numbers refer to pages on which a reference is made to an author or work of an author.

Italic numbers refer to pages on which a complete reference to a work by the author is given.

Boldface numbers indicate the first page of the articles in the book.

Abraham, R., 140,151, 399 Aleksandrian, R. A., 325, 325 Andreotti, A., 292, 295, 302, 303, 304, 305 Arnowitt, R„ 226, 249, 261, 409, 414 Aronszajn, N., 288, 295 Atiyah, M. F., 4, 10, 11, 12, 15, 134, 151, 265,

279 Aubin, T., 120, 722, 154,169, 306, 312, 318 Avez, A., 165,770,211,276 Azencott, R., 216, 276

Balian, R., 131, 757 Barbosa, J. L., 127 Bass, R. W., 407, 414 Baum, Paul, 3, 14, 75, 35, 378 Baxendale, P., 215, 276 Benabdallah, A., 150, 757 Berger, Marcel, 67, 80, 120, 722,129,130, 131,

134, 137, 138, 139, 150, 757, 187, 190, 193, 193, 195, 203, 227, 229, 235, 239, 246, 267, 273,279,281,284, 317, 318,318, 352,353

Berger, Melvyn S., 153, 156, 158, 162, 769, 170, 310, 318

Bers, L., 186,193, 293, 295, 296 Berthelot, P., 12, 75 Bishop, E., 81, 83, 86, 87, 289, 296 Bishop, R. L., 35, 41, 114, 116, 722, 174, 184,

190,193

Blaschke, W., 18, 24 Bloch, C , 131, 757 Bloch, S., 118, 722 Bochner, S., 17, 18, 24, 116, 722, 292, 296,

314, 319, 365, 368, 371 Borel, A., 75, 117, 722, 346, 347 Bott, R., 5, 10, 11, 12, 14, 75, 91, 92, 94, 95,

97, 134, 757, 265, 279, 375, 378 Bourguignon, J.-P., 230, 263, 314, 318 Brieskorn, E., 119, 722 Brill, D. R., 222, 245, 246, 257, 267, 410, 414 Burago, Ju. D., 187,795

Cahn, R. S., 139, 757 Calabi, E., 17,17,18,24,178,184,315,318,399 Cantor, M., 171, 171,184 Carlson, James, 27, 31, 52, 64 do Carmo, M., 127, 337, 341, 342 Cartan, E., 109, 772 Cartan, H., 105,108 Cartier,P.,211,216,276 Chavel, I., 191,795,351 Chazarain, J., 129, 757, 207, 209 Cheeger, J., 35, 37, 41, 187, 795, 352, 353, 375,

378 Cheng, Shiu-Yuen, 185, 186, 191, 795 Chern, S. S., 5, 75,30,57,44,64,154, 769,202,

203, 288, 296, 337

433

434 AUTHOR INDEX

Choquet-Bruhat, Y., 222, 248, 256, 261, 262, 409,4X1,414,411,422,423

Colin de Verdiere, Y., 129, 143, 146, 148, 149, 150,151,151,152,201,209

Combet, E., 151, 152 Cordes, H. O., 288, 296 Cotsaftis, M., 207, 209 Courant, R., 339, 340 Co wen, M., 44, 64 Crittenden, R., 174, 184, 190, 193

Day, M. M., 214, 216 D'Eath, P., 248, 263 Debney, George, 385, 386, 388, 389, 392, 393,

394 Deser, S., 222, 226, 248, 249, 257, 261, 262,

409, 410, 414 Dolbeault, P., 4, 15 Donnelly, Harold, 195 Duistermaat, H., 129, 152 Duistermaat, J. J., 205, 205, 206, 209 Duren, P. L., 363, 365, 366, 368, 370, 371 Dynkin, E. B., 214, 216

Ebin, D., 226, 227, 228, 229, 230, 232, 235, 236, 239, 246, 261, 262, 317, 318, 318

Eddington, A. S., 386, 394 Eells, James, 211, 212, 213, 216, 217, 343, 344 Eisenhart, L., 164,170 Eisenman (Pelles), D., 63, 64 Eliasson, H., 164, 170, 312, 314, 318 Ellis, G. F. R., 425, 429, 429 Elworthy, K. D., 212, 213, 217

Faris, W. G., 214, 217 Federer, H., 293, 296 Feldman, E., 191, 795,351 Fischer, Arthur E., 219, 226, 231,234, 235, 243,

247, 248, 249, 252, 258, 261, 262, 315, 318 Flaherty, F . J., 395 Flaschel, P., 140, 152, 208, 209 Flett, T. M., 367, 371 Frankel,T. T., 114, 115, 122 Frazier, Arlene P., 370,371 Freudenthal, H., 377, 378 Fujimoto, H., 28, 31 Furstenberg, H., 211, 216, 217 Furuya, S., 328, 337

Gauduchon, Paul, 134,137,138,139,150,151, 187, 190, 193,193, 195, 203, 352, 353

Geroch, Robert, 248, 261, 262, 401, 411, 414

Gieseker, D., 118,122

Gilkey, Peter B., 134, 139, 152, 195, 196, 203, 265, 267, 273, 277, 279, 279, 280, 281, 281, 283,284, 284, 285

Gluck, H., 310, 318 Goldberg, J. N., 387, 394 Goldberg, S. I., 25, 113, 114, 115, 121, 122,

284, 285 Grauert, H., 34, 41, 288, 292, 295, 296 Green, Leon W., 355 Green, Mark L., 27, 28, 31, 62, 64 Greene, R. E., 33, 33, 35, 36, 37, 38, 39, 41,

287, 290, 292, 295,296 Greenfield, S. J., 82, 83, 84, 85, 86, 87, 87 Griffiths, Phillip A., 27, 31, 43, 44, 52, 64, 96,

97,99,704,115,117,118,722 Gromoll,D., 35, 37,41,399 Gromov, M. L., 290, 296 Grothendieck, A., 9, 12,15,117, 722 Guillemin, V. W., 129, 752, 205, 297

Hahn, Kyong T., 364, 365, 370, 371 Hardy, G.H. , 366,370,371 Harish-Chandra, 358, 360 Hartshorne, R., 115,116, 722 Harvey, F. Reese, 84, 85, 87 Hashizume, M., 360,360 Hawking, S. W., 411, 414, 425, 428, 429,

429 Helgason, Sigurdur, 357, 357, 358, 359, 360,

399 Hermann, R., 378 Hersch, J., 352, 353 Hilbert, D., 156, 769, 339, 340 Hill, C. Denson, 301, 302, 303, 304, 305 Hironaka, H., 8, 10, 75 Hirschman, I. I., 214, 277 Hirzebruch, F., 3, 16, 114, 117, 119, 722,

161, 769, 346, 347 Hitchin, Nigel, 65,159, 769,312,318 Hormander, L., 34, 35, 38,41, 81,87, 205, 206,

208, 209, 294, 296, 297, 300 Howard, A., 115, 122,722 Hsiang, W. Y.; 337 Hua, L. K., 364,365, 366, 367, 370, J77 Huber,H.,150,752 Hunt ,L .R. ,81 ,85 ,S7

Iitaka, S., 80, 80 Illusie, L., 12, 75 Isaacson, R. A., 422, 423 Ishihara, S., 106, 108 Ishihara, T., 25 Ito, K., 215, 216, 277

AUTHOR INDEX 435

Jacobowitz, H., 307 Jensen, G., 318 John, F., 293, 295, 296, 298, 300 Johnson, K., 358, 359, 360

Kac, M., 130, 135, 152 Kakutani, S., 215, 217 Kato, T., 316, 319 Kazdan, Jerry L., 164, 170, 222, 231, 232, 245,

262, 290, 296, 309, 309, 310, 311, 312, 314, 319

Kemeny, J. G., 214, 217 Kerr, R. P., 386, 388, 394 Kesten,H., 211,214,217 Kiernan, Peter, 50, 64, 89 King, James R., 44, 64, 91, 91, 93, 96, 97 Kinnersley, W., 390, 394 Kitaoka, Y., 133, 152 Klingenberg, W., 121, 123, 140, 152, 208, 209,

399 Knapp, A. W., 214, 217 Kobayashi, S., 41, 44, 50, 64, 67, 80, 100, 101,

102, 104, 108, 113, 114, 115, 117, 118, 119, 120, 121, 122, 122, 123, 154, 159, 160, 169, 113,184,337,315,378,399

Kodaira, K., 40, 41, 68, 72, 80, 80, 114, 115, \\9,122,123,162,169

Koranyi, A., 363, 371 Kostant, B., 358, 360 Kowata, A., 360, 360 Kruskal, M. D., 427, 429 Kwack, Myung H., 99

Lai, Hon-Fei, 361, 361, 362 Landau, L. D., 416, 417, 423, 423 Lang, S., 235, 262 Lawruk, Bohdan, 321 Lawson, H. Blaine, Jr., 87, 337 Lax, P. D., 292, 296 Lazutkin, V. F., 131, 139, 152 LePotier, J., 118, 123 Leray, J., 91, 97 Lewy, H., 302, 305 Lichnerowicz, A., 228, 229, 232, 238, 239, 240,

256, 262, 312, 315, 319, 416, 423 Lifshitz, E. M., 416, 417, 423,423 Little wood, J. E., 366, 370, 371 Look, K. H., 365, 366, 367, 371 Louchard,G., 131,152 Lu, R.-Q., 363, 371 Ludwig, D., 297, 300

MacCallum, M. A. H., 420,421, 422,423

MacPherson, R., 15, 16 Madore, J., 422, 423 Maeda, M., 330, 337 Malgrange, B., 288, 292, 294, 296 Malliavin, P., 216, 217 Malyutov, M. B., 214, 216 Markus, L., 399 Marsden, Jerrold E., 219, 226, 231, 234, 236,

243, 248, 249, 252, 258, 261, 262 Martin, W. T., 11,24 Matsushima, Y., 120, 123, 365, 371 Mazet, Edmond, 134, 137, 138, 139, 150, 151,

187, 190, 193,193, 195, 203, 352, 353 McKean, H. P., Jr., 130,131,138,150,152,192,

193,193,195,200,203,215,217,273,280, 282, 284, 285

Meyer, W., 35, 41, 399 Millman, Richard S., 105,105,106,107,108 Milnor, J., 133, 140,152, 211, 217 Minemura, K., 360, 360 Misner, C. W., 226, 249, 261, 409, 414 Mitchell, Josephine, 363, 364, 365, 366, 370,

371 Moncrief, V., 248, 263 Morimoto, A., 107, 108 Morrey, C. B., 288, 296 Morrow, J., 72, 80, 162,169 Moser, Jurgen, 39, 41, 109, 158, 170, 310, 316,

319 Mumford, D., 160, 166,169 Muto, Y., 263 Myers, S. B., 19,24, 114,725

Naimark, M. A., 130, 152 Nakai, M., 40, 41 Nakai, Y., 68, 80 Narasimhan, R., 289, 294, 296 Nelson, E., 228, 262 Newman, E. T., 386, 394 Nirenberg, L., 87, 171, 172, 173, 179, 182,

184, 231, 262 Nirenberg, R., 82, 86, 87 Nomizu, K., 108, 121, 122, 123, 173, 184, 291,

296, 399

Obata, Morio, 231, 262, 373 Ochiai, Takushiro, 50, 64, 113, 114, 115, 117,

118, 119, 120,123 Okamoto, K., 360, 360 O'Murchadha, N., 256, 263 O'Neill, B., 35, 41 Otsuki, Tominosuke, 327, 327, 328, 329, 330,

331, 332, 334, 335, 337, 337

436 AUTHOR INDEX

Ozeki,H., 291, 296 Ozols, V., 375, 376, 377, 378

Patodi, V. K., 134, 139, 151, 152, 195, 197, 198,199, 200, 203, 265, 278, 279, 280

Payne, L. E., 131,152,187,193 Penrose, R., 399, 399, 425, 428, 429 Pirani, F. A. E., 417, 423 Pleijel, A., 186, 193 Pogorelov, A. V., 18, 24 Poincare, H., 109, 112, 154, 166, 169, 352,

353 Polya, G., 187,193 Protter, M. H., 19,20,24,288,296

Quillen, D., 12, 16

Ray, D. B., 343, 344, 347 Reid, W. T., 352, 353 Remmert, R., 117, 122, 289, 296 Robin, L., 217 Robinson, I., 387, 390, 394 Romberg, B. W., 363, 366, 370, 371 Rosenblatt, M., 214, 217 Royden, H. L., 50, 64, 292, 296 Ruse, H. S., 387,394

Sachs, R. K., 386, 387,394 Sacks, Jonathan, 265, 279, 280, 281 Safarevic, I. R., 66, 68, 80, 115,123 Sakai, T., 138, 139, 152 Samelson, H., 343, 347 Sampson, J. H., 341, 342 Sario, L., 40, 41 Schaeffer, David G., 297, 339, 340, 340 Schechter, M., 293, 295, 296 Schild, A., 386, 387, 388, 394 Schmid, W., 44, 64 Schmidt, B., 413, 414 Seeley, R. T., 266, 280, 282, 285 Segal, G. B., 12, 15, 378 Serre, J.-P., 15 Shapiro, A., 10, 75 Shields, A. L., 363, 366, 370, 371 Sibner, L., 96, 97 Sibner, R., 96, 97 de Siebenthal, J., 376, 378 Siegel, C. L., 40,41 Singer, I. M., 4, 15, 129, 134, 138, 152, 193,

193, 195, 200, 203, 273, 280, 282, 284, 285, 343, 344, 347

Siu, Y.-T., 288, 292, 296 Smith, J. W., 399

Smith, R .T . , 341, 341, 342 Smyth, B., 115, 122 Snell, J. L., 214, 217 Sobolev, S. L., 325, 325 Spitzer, F., 214, 217 Stanton, Nancy K., 343, 344, 347 Stein, K., 81, 87 Struik, D., 299, 300 Suwa, T., 80, 80 Svarc, A. S., 211, 217 Szego, G., 187,193

Tanaka, N., 109, 112 Tanno, S., 139, 152 Taub, A. H., 415, 420, 421, 422, 423, 423 Thorpe, John A., 425 Toda, N., 28, 31 Toledo, D., 96, 97 Tong, Y.-L. L., 96, 97 Trautman, A., 390, 394 Trudinger, N. S., 164,170, 314, 319 Tsukamoto, Y., 67, 80

Urabe, M., 328, 337

Vincent, G., 378 Virtanen, K. I., 40, 41

Waechter, R. T., 131, 152 Walker, H., 231, 262 Wallach, N. R., 341, 342, 358, 359, 360 Wang, H. C , 117, 123 Warner, F. W., 131, 152, 164, 170, 222, 231,

232, 245, 262, 290, 296, 309, 309, 310, 311, 312, 314, 319

Watson, G. N., 352, 353 Weinberger, H. F., 19, 20, 24, 187,193 Weinstein, A., 129, 152, 207, 209, 322, 323,

325 Wells, R. O., Jr., 64, 81, 81, 82, 83, 84, 85, 86,

87, 87, 88 Weyl, H., 365, 371 Whitehead, G. W., 10, 16 Whitham, G. B., 420, 423 Widder, D. V., 214, 217 Wiener, N., 212, 217 Willmore, T. J., 19, 24, 315, 319 Witten, L., 390, 394, 407, 414 Wolf, Joseph A., 235, 236, 237, 247, 262, 263,

315, 318, 363, 371, 377, 378, 379, 399 Wu, H., 33, 33, 35, 36, 37, 38, 39, 41, 67, 80,

99,104, 287, 290, 292, 295, 296 Wu, W. T., 361, 362

AUTHOR INDEX 437

Yamabe, H., 164, 170, 314, 319 Zalgaller, V. A., 187, 193 Yano, K., 106, 108, 236, 263, 314, 319 Zeeman, E. C , 398 Yau, S. T., 154, 159, 163, 164, 166,169 Zund, J. D., 393, 394 York, J. W., 256, 263 Zygmund, A., 367, 371

SUBJECT INDEX

A*\ 137 A*\ 137 A*B, 137 affine group, 236 affine transformation, 236 algebraic classification of curvature tensors,

417 amenable, 214 ample bundle, 115 associated generic submanifold, 83 asymptotic expansion, 138 asymptotic invariant, 266 averaged Lagrangian, 420

bu, 141 back-reaction, 422 Bianchi identity, 390 Bieberbach theorem, 235 Bishop family of analytic discs, 86 blowing up, 72 Borel fibre space, 379 Borel structure, 379 boundary cohomology, 303 bump function, 178

C+, 142 C*(£"), 172 Ck+9(E), 172 Calabi conjecture, 66, 69, 315 canonical decomposition, 228 canonical measure, 132 canonical relation, 206 Cartan structure equations, 390 Cauchy problem, 303 causal geodesic, 428

chains, 111, 112 Chern form, 93 Clifford bundle, 275 Clifford translation, 377 cohomology class of Keller-Maslov-Arnol'd,

208 cohomology ring of/22n>2, 361 cohomology vanishing theorems, 40 complete metric, 310 complete Riemannian manifold, 314 complex Laplacian, 139 complex null tetrad, 392 conformal deformation, 310 conformal method, 256 conformal tensor, 387 conformally equivalent, 233 conformorphism, 373 conjugate points, 397 constant holomorphic sectional curvature,

265, 281 constant scalar curvature, 265 constant sectional curvature, 265 constraint

divergence, 249, 257 Hamiltonian, 221, 249 nonlinear, 225

constraint equation, 249, 256 linearized, 225

constraint set, 259 Cousin problem, 105 Cousin structure, 106 C/?-function, 83 Cotangent vector, 82 critical point, 244, 252, 253 curvature, 351

439

440 SUBJECT INDEX

constant holomorphic sectional, 265, 281 constant scalar, 265 constant sectional, 265 flat, 265 Gaussian, 309 holomorphic bisectional, 113 positive holomorphic sectional, 72 positive Ricci, 68, 72 positive scalar, 65, 67, 77, 79 Ricci, 219, 313 scalar, 288, 290, 311 zero scalar, 314

curvature tensors, algebraic classification of, 417

cuspidal parabolic subgroup, 381

D», 146 a problem, 38 dK, 130 deformation

conformal, 310 finite, 220 first order, 220 geometric, 228 infinitesimal, 220, 248 nonintegrable, 255

J, 130, 132 J2, 134, 143 A\ 139 difference formula, 381 Dirichlet norm, 40 distribution character, 381 divergence, 227 divergence constraint, 249, 257 dual lattice, 133

£", 140 E, 141 Eu, 141 eigenvalue, 312, 351 Einstein, 265 Einstein equation, 17, 219, 406 Einstein-Maxwell theory, 385 Einstein metric, 17, 234 Einstein tensor, 385 electromagnetic field tensor, 385 elliptic equation, nonlinear, 310 elliptic operator, 230, 266, 287 elliptic pseudodifferential operator, 205 energy function, 140 envelope, 299 exceptionality, 82 extendible, 81

F, 134, 143 /(*, 0 , 142 /-contact Lie group, 107 /-Lie group, 107 /-manifold, 106 /-structure, 105 field equations, 261 finite deformation, 220 first order deformation, 220 first variation, 312 fixed membrane problem, 185 flat curvature, 265 flat Riemannian metric, 221, 234 flat torus, 133 Fourier integral operator, 206, 297 free boundary value problem, 339 free membrane problem, 185 FSCHE, 144 FSRHE, 134 fundamental solution of the complex heat

equation, 143 fundamental solution of the real heat equation,

129 Furuya's inequality, 328 future complete, 396 future 1-connected, 395 future-timelike curve, 395 future-timelike vector, 395

G, 136 G , 140 G , 140 Gauss-Bonnet theorem, 309 Gaussian curvature, 309 Gaussian measure, 215 general relativity, 248,401 generalisation of the weak gravitational wave,

417 generalized geodesic flow, 355 generic, 81, 82 generic submanifold, associated, 83 geodesic and shearfree null vector field, 386,

387 geodesically convex, 35 geometric deformation, 228 goemtric optics, 417 geometric realization, 381 Grassmannian, 27 gravitational energy, 408 gravitational waves, 415, 417

generalisation of weak, 417 high frequency, 417

group of diffeomorphisms, 227

SUBJECT INDEX 441

Hki 137 h, 141 Hamiltonian constraint, 221, 249 Hamiltonian density, 251 harmonic function, 288 harmonic mapping, 25 heat density, 212 Hermitian metric, 289 Hessian, 228, 244, 252, 253 high frequency gravitational wave, 417 Hodge filtration, 96 Holder norm, 172 holomorphic bisectional curvature, 113 holomorphic equivalence, 109 holomorphic form, 36 holomorphic function, 36 holomorphic mapping, 27 holomorphic itoorsion, 343 holomorphic set, 85

locally, 85 holomorphically convex, 85

locally, 85 holonomy group, 237 homogeneous fibre bundle, 221 homogeneous space, 236 horocycle field, 355 horosphere, 355

/„+, 145 In, 145 index, 140 induced representation, 379 infinitesimal deformation, 220, 248 injectivity radius, 131 instability, 225 intermediate volume element, 25 intersection, 94 irreducible, 214 isolated solution, 244, 257 isolated subset of solutions, 225 isometric, 131 isometry group, 236 isospectral deformation, 133

7'L, 140 Jw, 140 Jacobi field, 397

K, 130 Kky 137 Kahler, 265 Kahler manifold, 33 Kahler metric, 289

Killing vector field, 388

se, 140 L>A(E\ 172 L2-adjoint, 224 A*, 133 *» 131 Laplace-de Rham operator, 228 Laplacian, 132, 265, 288, 311, 351

complex, 139 Lichnerowicz, 238, 239, 312 rough, 228

Lefschetz fixed point formula, 91 Levi form, 84, 292, 302 Lewy example, 302 Lichnerowicz equation, 256 Lichnerowicz Laplacian, 238, 239, 312 Lie derivative, 227 linearization stable, 219, 220 linearized constraint equation, 225 linearized equation, 220 linearized evolution equation, 225 locally CR, 81 locally CR-submanifold, 82 locally holomorphic set, 85 locally holomorphically convex, 85 locally solvable, 301

not, 301 Lorentz 4-manifold, 391 Lorentz manifold, 387, 395 Lorentz metric, 260

M0", 141 M2, 136 Maxwell's equations, 356 Mayer-Vietoris sequence, 303 measurable family of C°° manifolds, 379 method of continuity, 298 metric

complete, 310 Einstein, 17, 234 flat Riemannian, 221, 234 Hermitian, 289 Kahler, 289 Lorentz, 260 Ricci flat, 234, 235, 315

metric on the wv-plane, 329 Minakshisundaram coefficient, 267 minimum principle, 185 Minkowski space, 222 mul(A), 132 multiplicity, 132

442 SUBJECT INDEX

N((xd), 147 nWi 140 Nijenhuis torsion tensor, 106 nodal domain, 186 nodal line, 186 nodal set, 186 nondegenerate, 140 nondegenerate critical submanifold, 140 nonelliptic complex, 302 nonintegrable deformation, 255 nonlinear constraint, 225 nonlinear differential equation, 327 nonlinear differential operator, 220 nonlinear elliptic equation, 310 nonlinear evolution equation, 225 nonsolvability examples, 302 norm, 172

Holder, 172 Sobolev, 172

normal form, 109, 112 not locally solvable, 301 v, 132 K"0, 132 null eigenvector field, 387 null vector, 395 nullity, 140

Ol 329 obstacle problem, 339 Q(M\ 140 orthogonal splitting, 230

Pn\ 145 /V, 145 partial differential equation, 219 partial spin structure, 379 partially C°° bundle, 379 partially harmonic spinor, 380 paving, 177 period T, 328 periodic geodesic, 131,134, 205 perturbation, 219 Petrov classification, 387 physical optics, 417 Picard theorem, 27 Plancherel measure, 381 plurisubharmonic, strictly, 34 Poincare dual, 93 Poincare map, 206 polynomial in z, 144 positive holomorphic sectional curvature, positive Ricci curvature, 68, 72

positive scalar curvature, 65, 67, 77, 79 projective variety, Todd class of, 7 proper mapping, 287 pull-back, 297 push-forward, 297

quasiconformal mapping, 25 quotient space, 258

(R«IAigJA\m Radon inversion formula, 298 Radon transform, 297 random walk, 213 rational surfaces, 66 iZC-singular point, 361 real analytic imbedding, 288 real analytic manifold, 288 real heat equation, 134

fundamental solution of the, 129 real projective plane, 351 recurrent, 214 reductive Lie group, 380 regularity theorem, 234 residue, 91 residue form of kernel type, 93 de Rham complex, 276 P, 131 Ricci curvature, 219, 313 Ricci flat metric, 234, 235, 315 Riemann-Hilbert problem, 303 Riemannian bundle, 171 Riemannian manifold, 131

complete, 314 Riemannian structure, 131 Riemannian Wiener measure, 211 rough Laplacian, 228

5,148 Sk, 136 saddle point, 253 scalar curvature, 288, 290, 311

zero, 314 scalar curvature equation, 219 Schauder ring property, 227 Schwarzschild model, 425 Schwarzschild radius, 426 second fundamental form, 248 second order condition, 223 second variation, 312 2*> 139 singularity, 340, 411

72 small displacement, 375 Sobolev inequality, 171, 293

SUBJECT INDEX 443

Sobolev norm, 172 Sobolev space, 226 Sobolev theorem, 226 spacelike hypersurface, 248 spacelike vector, 395 spacetime, 221 space-time plane, 396 Spec, 130, 133 Spec(/0, 130 Spec(M,^), 132 special symplectic space, 323, 324 spectral invariant, 278 spectrum, 85, 130, 132, 266

of positive elliptic operators, 205 stable, 219

linearization, 219, 220 stable minimal surface, 127 standard sphere, 231, 243 Stein manifold, 33, 289 stochastic matrix, 214 stochastic parallel transport, 216 stress-energy tensor, 385 strictly plurisubharmonic, 34 subharmonic function, 37 subprincipal symbol, 205 symmetric space, 380 symmetric tensor fields, 227

tangent vector, 395 tangential Cauchy-Riemann complex, 302 tautological line bundle, 115 tensor density, 249 timelike vector, 395 time-orientable, 395 Todd class of a projective variety, 7 transient, 214 transverse, 255

tubular domain, 17 two-timing method, 417, 418 type r „ 142

U, 144 Ui9 137 ut, 136 umbilical point, 112 uniformization theorem, 233 unique continuation property, 287 Urabe's conjecture, 328

V„ 137 vanishing theorem of Malgrange and Siu, 288 volume, 132 Volume(A/,^), 132

Wu H9 wave-front, 297 weak gravitational field in vacuo, 416 weak gravitational wave, generalisation of,

417 Weyl conformal curvature tensor, 392 WFf 297 Whitham's technique, 421 Whitney imbedding theorem, 289

Xny 140

Yamabe's question, 314

Z(z), 144 za, 142 zero scalar curvature, 314

= F, 142 = F» 142 •**, 137

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