Seminar on Differential Geometry. (AM-102) ( Annals of Mathematics Studies )

Publication series ：Annals of Mathematics Studies

Author： Yau Shing-tung  

Publisher： Princeton University Press‎

Publication year： 2016

E-ISBN: 9781400881918

P-ISBN(Paperback): 9780691082684

Subject： O186.1 differential geometry

Keyword： 几何、拓扑,数学

Language： ENG

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Description

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.

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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

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