Yau Seminar on Differential Geometry

1. Auflage 2016
ISBN: 978-1-4008-8191-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 102, 720 Seiten

Reihe: Annals of Mathematics Studies

This collection of papers constitutes a wide-ranging survey of recent developments in differential geometry and its interactions with other fields, especially partial differential equations and mathematical physics. This area of mathematics was the subject of a special program at the Institute for Advanced Study in Princeton during the academic year 1979-1980; the papers in this volume were contributed by the speakers in the sequence of seminars organized by Shing-Tung Yau for this program. Both survey articles and articles presenting new results are included. The articles on differential geometry and partial differential equations include a general survey article by the editor on the relationship of the two fields and more specialized articles on topics including harmonic mappings, isoperimetric and Poincaré inequalities, metrics with specified curvature properties, the Monge-Arnpere equation, L2 harmonic forms and cohomology, manifolds of positive curvature, isometric embedding, and Kraumlhler manifolds and metrics. The articles on differential geometry and mathematical physics cover such topics as renormalization, instantons, gauge fields and the Yang-Mills equation, nonlinear evolution equations, incompleteness of space-times, black holes, and quantum gravity. A feature of special interest is the inclusion of a list of more than one hundred unsolved research problems compiled by the editor with comments and bibliographical information.

Autoren/Hrsg.

Yau, Shing-Tung

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie

Inhaltsverzeichnis

Frontmatter, pg. i
TABLE OF CONTENTS, pg. v
INTRODUCTION, pg. ix
SURVEY ON PARTIAL DIFFERENTIAL EQUATIONS IN 3 DIFFERENTIAL GEOMETRY, pg. 1
POINCARÉ INEQUALITIES ON RIEMANNIAN MANIFOLDS, pg. 73
BONNESEN-TYPE INEQUALITIES IN ALGEBRAIC GEOMETRY, I: INTRODUCTION TO THE PROBLEM, pg. 85
LIOUVILLE THEOREMS FOR HARMONIC MAPPINGS, AND AN APPROACH TO BERNSTEIN THEOREMS, pg. 107
SUBHARMONIC FUNCTIONS, HARMONIC MAPPINGS AND ISOMETRIC IMMERSIONS, pg. 133
AN ISOPERIMETRIC INEQUALITY AND WIEDERSEHEN MANIFOLDS, pg. 143
ON THE BLASCHKE CONJECTURE, pg. 159
BEST CONSTANTS IN THE SOBOLEV IMBEDDING THEOREM THE Y AMABE PROBLEM, pg. 173
GAUSSIAN AND SCALAR CURVATURE, AN UPDATE, pg. 185
CONFORMAL METRICS WITH ZERO SCALAR CURVATURE AND A SYMMETRIZATION PROCESS VIA MAXIMUM PRINCIPLE, pg. 193
RIGIDITY OF POSITIVELY CURVED MANIFOLDS WITH LARGE DIAMETER, pg. 203
COMPLETE THREE DIMENSIONAL MANIFOLDS WITH POSITIVE RICCI CURVATURE AND SCALAR CURVATURE, pg. 209
ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT MEAN CURVATURE IN MINKOWSKI SPACE, pg. 229
APPLICATIONS OF THE MONGE-AMPERE OPERATORS TO THE DIRICHLET PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS, pg. 239
EXTREMAL KÄHLER METRICS, pg. 259
L2 HARMONIC FORMS ON COMPLETE MANIFOLDS, pg. 291
L2-COHOMOLOGY AND INTERSECTION HOMOLOGY OF SINGULAR ALGEBRAIC VARIETIES, pg. 303
FUNCTION THEORY OF NONCOMPACT KÄHLER MANIFOLDS OF NONPOSITIVE CURVATURE, pg. 341
KÄHLER MANIFOLDS WITH VANISHING FIRST CHERN CLASS, pg. 359
COMPACTIFICATION OF NEGATIVELY CURVED COMPLETE KAHLER MANIFOLDS OF FINITE VOLUME, pg. 363
LOCAL ISOMETRIC EMBEDDINGS, pg. 381
YANG-MILLS THEORY: ITS PHYSICAL ORIGINS AND DIFFERENTIAL GEOMETRIC ASPECTS, pg. 395
SYMMETRY AND ISOLATED SINGULARITIES OF CONFORMALLY FLAT METRICS AND OF SOLUTIONS OF THE YANG-MILLS EQUATIONS, pg. 423
ON PARALLEL YANG-MILLS FIELDS, pg. 443
VARIATIONAL PROBLEMS FOR GAUGE FIELDS, pg. 455
SOME GEOMETRICAL ASPECTS OF INTEGRABLE NONLINEAR EVOLUTION EQUATIONS, pg. 465
THE CONFORMALLY INVARIANT LAPLACIAN AND THE INSTANTON VANISHING THEOREM, pg. 483
CAUSALLY DISCONNECTING SETS, MAXIMAL GEODESICS AND GEODESIC INCOMPLET-ENESS FOR STRONGLY CAUSAL SPACE-TIMES, pg. 499
RENORMALIZATION, pg. 507
METRICS WITH PRESCRIBED RICCI CURVATURE, pg. 525
BLACK HOLE UNIQUENESS THEOREMS IN CLASSICAL AND QUANTUM GRAVITY, pg. 539
GRAVITATIONAL INSTANTONS, pg. 603
SOME UNSOLVED PROBLEMS IN CLASSICAL GENERAL RELATIVITY, pg. 631
PROBLEM SECTION, pg. 669
Backmatter, pg. 707