Contemporary Geometry

Early one morning in April of 1987, the Chinese mathematician J. -Q. Zhong died unexpectedly of a heart attack in New York. He was then near the end of a one-year visit in the United States. When news of his death reached his Chinese-American friends, it was immediately decided by one and all that something should be done to preserve his memory. The present volume is an outgrowth of this sentiment. His friends in China have also established a Zhong Jia-Qing Memorial Fund, which has since twice awarded the Zhong Jia-Qing prizes for Chinese mathematics graduate students. It is hoped that at least part of the reasons for the esteem and affection in which he was held by all who knew him would come through in the succeeding pages of this volume. The three survey chapters by Li and Treibergs, Lu, and Siu (Chapters 1-3) all center around the areas of mathematics in which Zhong made noteworthy contributions. In addition to putting Zhong's mathematical contributions in perspective, these articles should be useful also to a large segment of the mathematical community; together they give a coherent picture of a sizable portion of contemporary geometry. The survey of Lu differs from the other two in that it gives a firsthand account of the work done in the People's Republic of China in several complex variables in the last four decades.

Contenu

Zhong Jia-Qing (1937-1987).- Publications of Zhong Jia-Qing.- I: Surveys of Contemporary Geometry.- 1. Applications of Eigenvalue Techniques to Geometry.- 1.1. Eigenvalues of the Laplacian.- 1.2. Upper Bounds via Conformal Geometry.- 1.3. Upper Bounds Using Cheng's Comparison Theorem.- 1.4. Lower Bounds by Gradient Estimates.- 1.5. Isoperimetric Inequalities.- 1.6. A Lower Bound of Choi and Wang and Its Application to Minimal Surfaces.- 1.7. Nodal Domains and Multiplicity Bounds.- 1.8. Stability of Minimal Surfaces.- References.- 2. The Theory of Functions of Several Complex Variables in China from 1949 to 1989.- 2.1. The Classical Domains.- 2.2. The Classical Manifolds.- 2.3. Homogeneous Manifolds.- 2.4. Integral Representations and Their Boundary Values.- 2.5. The Schwarz Lemma.- 2.6. Harmonic Analysis in Classical Domains and on Their Characteristic Manifolds.- 2.7. Pseudoconvex Domains.- References.- 3. Uniformization in Several Complex Variables.- 3.1. Characterization of Compact Type by Topological Conditions.- 3.2. Characterization of Noncompact Type by Topological Conditions.- 3.3. Characterization of Compact Type by Curvature Conditions.- 3.4. Characterization of Noncompact Type by Curvature Conditions.- 3.5. Characterization of Euclidean Space by Curvature Conditions.- 3.6. Characterization of Noncompact Type by Compact Quotients.- 3.7. Metric Rigidity.- References.- II: Selected Papers of Zhong Jia-Qing.- 4. Harmonic Analysis on Rotation Groups: Abel Summability.- 5. Coxeter-Killing Transformations of Simple Lie Algebras.- 6. Dimensions of the Rings of Invariant Differential Operators on Bounded Homogeneous Domains.- 7. On Prime Ideals of the Ring of Differential Operators.- 8. On the Sum of Class Functions of Weyl Groups.- 9. The Trace Formula of the Weyl Group Representations of the Symmetric Group.- 10. Some Types of Nonsymmetric Homogeneous Domains.- 11. The Extension Spaces of Nonsymmetric Classical Domains.- 12. Cohomology of Extension Spaces for Classical Domains.- 13. The Degree of Strong Nondegeneracy of the Bisectional Curvature of Exceptional Bounded Symmetric Domains.- 14. On the Estimate of the First Eigenvalue of a Compact Riemannian Manifold.- 15. Curvature Characterization of Compact Hermitian Symmetric Spaces.- 16. Schubert Calculus and Schur Functions.- 17. An Expansion in Schur Functions and Its Applications in Enumerative Geometry.