The Yamabe problem

1987/01/01 by John M. Lee, Thomas H. Parker · 1,183 citations
Mathematics · #Spectral Theory in Mathematical Physics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Yamabe flow #Mathematics #Geometry #Scalar curvature #Sectional curvature #Curvature

paper · pdf · doi:10.1090/s0273-0979-1987-15514-5

published in Bulletin of the American Mathematical Society 17(1), 37-91 (American Mathematical Society)

openalex publication_date 1987/01/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Contents 1. Introduction 2. Geometric and analytic preliminaries 3. The model case: the sphere 4. The variational approach 5. Conformai normal coordinates 6. Stereographic projections 7. The test function estimate 8. General relativity 9. Analysis on asymptotically flat manifolds 10. The positive mass theorem 11. Solution of the Yamabe problem Appendix: Witten's proof Bibliography

Citations

• Elliptic Partial Differential Equations of Second Order
• Conformally flat manifolds, Kleinian groups and scalar curvature
• On the proof of the positive mass conjecture in general relativity
• The mass of an asymptotically flat manifold
• A relation between pointwise convergence of functions and convergence of functionals
• A new proof of the positive energy theorem
• Dynamical Structure and Definition of Energy in General Relativity

Cited by

• Initial data for fluid bodies in general relativity
• On the Bartnik extension problem for the static vacuum Einstein equations
• The curvature of contact structures on 3–manifolds
• Mapping superintegrable quantum mechanics to resonant spacetimes
• Examples of non-isolated blow-up for perturbations of the scalar\n curvature equation on non locally conformally flat manifolds
• Local gravity theories in conformal superspace
• The Weyl tensor of gradient Ricci solitons
• Metrics of Negative Ricci Curvature
• Asymptotic expansions and conformal covariance of the mass of conformal differential operators
• The trace and the mass of subcritical GJMS operators
• A positive mass theorem in three dimensional Cauchy–Riemann geometry
• A remark on the rigidity of Poincaré–Einstein manifolds
• A Mass for ALF Manifolds
• Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature
• Boundary definition of a multiverse measure
• The Yamabe Constant on Noncompact Manifolds
• Asymptotic Gluing of Asymptotically Hyperbolic Solutions to the Einstein Constraint Equations
• Is general relativity ‘essentially understood’?
• Yamabe flow with prescribed scalar curvature
• Metrics of constant scalar curvature on sphere bundles
• Yang-Mills Detour Complexes and Conformal Geometry
• A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends
• Harmonic coordinates [wikipedia]
• Scalar curvature [wikipedia]
• Yamabe problem [wikipedia]