Review of geometry and analysis

2000/01/01 by Shing–Tung Yau · 5 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory #Mathematics #Geometry

paper · doi:10.4310/ajm.2000.v4.n1.a16

openalex publication_date 2000/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

In this article, we shall discuss what the author considers to be important in geometry and related subjects.Since the time of the Greek mathematicians, geometry has always been in the center of science.Scientists cannot resist explaining natural phenomena in terms of the language of geometry.Indeed, it is reasonable to consider geometric objects as parts of nature.Practically all elegant theorems in geometry have found applications in classical or modern physics.In order to understand the future of geometry, it is perhaps useful to review what was known in the past.Clearly what I consider to be important may not be viewed to be so by others.Also, we should always keep in mind that what is fashionable now may not be so tomorrow.A theory can be judged to be successful only if its consequences help us understand the basic structure or the beauty of geometry.While we shall divide the subject into several categories, the division is artificial, as the development of each section depends on other sections heavily.

Citations

• Four-manifolds with positive isotropic curvature
• Special Lagrangian Fibrations II: Geometry
• Uniformization of four manifolds
• Topological sigma model and Donaldson-type invariants in Gromov theory
• Special Lagrangian Fibrations I: Topology
• Algebraic surfaces and Seiberg-Witten invariants
• Attractors and Arithmetic
• Family Blow Up Formula, Admissible Graphs and the counting of Nodal Curves
• Comparison of the algebraic and the symplectic Gromov-Witten invariants
• On the work of Givental relative to mirror symmetry
• The moduli space of 3-folds withK=0 may nevertheless be irreducible
• A Method for Generating Solutions of Einstein's Equations
• Heat Equations on Minimal Submanifolds and Their Applications
• A note on our previous paper: On the existence of Hermitian Yang–Mills connections in stable vector bundles
• Complete affine hypersurfaces. Part I. The completeness of affine metrics
• On the proof of the positive mass conjecture in general relativity
• A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
• On Einstein Manifolds of Positive Sectional Curvature
• Boundary regularity and embedded solutions for the oriented Plateau problem
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• Equivariant Gromov - Witten Invariants
• The enumerative geometry of K3 surfaces and modular forms
• Calibrated geometries

Cited by

• Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity
• Calabi-Yau Conjecture for Minimal Hypersurfaces in ℝ⁴ with bounded geometry
• NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY
• Scalar curvature [wikipedia]
• Shing-Tung Yau [wikipedia]