Seminar on Differential Geometry

Shing-Tung Yau, Shing-tung Yau

• Publisher: Princeton University Press
• Year: 2016
• Print ISBN: 9780691082967, 9780691082684
• eBook ISBN: 9781400881918
• Topic: Mathematics
• Subtopic: Differential Geometry
• Index: Mathematics

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.

Table of contents

• Cover
• Title
• Copyright
• TABLE OF CONTENTS
• INTRODUCTION
• SURVEY ON PARTIAL DIFFERENTIAL EQUATIONS IN DIFFERENTIAL GEOMETRY
• POINCARÉ INEQUALITIES ON RIEMANNIAN MANIFOLDS
• BONNESEN-TYPE INEQUALITIES IN ALGEBRAIC GEOMETRY, I. INTRODUCTION TO THE PROBLEM
• LIOUVILLE THEOREMS FOR HARMONIC MAPPINGS, AND AN APPROACH TO BERNSTEIN THEOREMS
• SUBHARMONIC FUNCTIONS, HARMONIC MAPPINGS AND ISOMETRIC IMMERSIONS
• AN ISOPERIMETRIC INEQUALITY AND WIEDERSEHEN MANIFOLDS
• ON THE BLASCHKE CONJECTURE
• BEST CONSTANTS IN THE SOBOLEV IMBEDDING THEOREM: THE YAMABE PROBLEM
• GAUSSIAN AND SCALAR CURVATURE, AN UPDATE
• CONFORMAL METRICS WITH ZERO SCALAR CURVATURE AND A SYMMETRIZATION PROCESS VIA MAXIMUM PRINCIPLE
• RIGIDITY OF POSITIVELY CURVED MANIFOLDS WITH LARGE DIAMETER
• COMPLETE THREE DIMENSIONAL MANIFOLDS WITH POSITIVE RICCI CURVATURE AND SCALAR CURVATURE
• ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT MEAN CURVATURE IN MINKOWSKI SPACE
• APPLICATIONS OF THE MONGE-AMPERE OPERATORS TO THE DIRICHLET PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS
• EXTREMAL KÄHLER METRICS
• L^2 HARMONIC FORMS ON COMPLETE MANIFOLDS
• L^2-COHOMOLOGY AND INTERSECTION HOMOLOGY OF SINGULAR ALGEBRAIC VARIETIES
• FUNCTION THEORY OF NONCOMPACT KÄHLER MANIFOLDS OF NONPOSITIVE CURVATURE
• KÄHLER MANIFOLDS WITH VANISHING FIRST CHERN CLASS
• COMPACTIFICATION OF NEGATIVELY CURVED COMPLETE KÄHLER MANIFOLDS OF FINITE VOLUME
• LOCAL ISOMETRIC EMBEDDINGS
• YANG-MILLS THEORY: ITS PHYSICAL ORIGINS AND DIFFERENTIAL GEOMETRIC ASPECTS
• SYMMETRY AND ISOLATED SINGULARITIES OF CONFORMALLY FLAT METRICS AND OF SOLUTIONS OF THE YANG-MILLS EQUATIONS
• ON PARALLEL YANG-MILLS FIELDS
• VARIATIONAL PROBLEMS FOR GAUGE FIELDS
• SOME GEOMETRICAL ASPECTS OF INTEGRABLE NONLINEAR EVOLUTION EQUATIONS
• THE CONFORMALLY INVARIANT LAPLACIAN AND THE INSTANTON VANISHING THEOREM
• CAUSALLY DISCONNECTING SETS, MAXIMAL GEODESICS AND GEODESIC INCOMPLETENESS FOR STRONGLY CAUSAL SPACE-TIMES
• RENORMALIZATION
• METRICS WITH PRESCRIBED RICCI CURVATURE
• BLACK HOLE UNIQUENESS THEOREMS IN CLASSICAL AND QUANTUM GRAVITY
• GRAVITATIONAL INSTANTONS
• SOME UNSOLVED PROBLEMS IN CLASSICAL GENERAL RELATIVITY
• PROBLEM SECTION