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The Yamabe problem

1987; American Mathematical Society; Volume: 17; Issue: 1 Linguagem: Inglês

10.1090/s0273-0979-1987-15514-5

1088-9485

John M. Lee, Thomas H. Parker,

Nonlinear Partial Differential Equations

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 Introduction.Riemannian differential geometry originated in attempts to generalize the highly successful theory of compact surfaces.From the earliest days, conformai changes of metric (multiplication of the metric by a positive function) have played an important role in surface theory.For example, one consequence of the famous uniformization theorem of complex analysis is the fact that every surface has a conformai metric of constant (Gaussian) curvature.This provides a "standard model" for each homeomorphism class of surfaces, and reduces topological questions to differential geometric ones.Life would be simple if the naive generalization of this theorem held in higher dimensions: every «-manifold would have a conformai metric of constant curvature, and questions in differential topology would be reduced to geometric questions about the constant-curvature models.However, it is easy to see that this cannot be true.In general the problem is highly overdetermined: the curvature tensor has on the order of n 4 independent components, while a conformai change of metric allows us to choose only one unknown function.For example, if n ^ 4, the Weyl tensor, formed from the components of the Riemannian curvature tensor, is conformally invariant and vanishes if and only if the metric is locally conformally equivalent to the Euclidean metric.From this point of view it seems natural instead to seek a