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The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. Yamabe (1960) claimed to have a solution, but Trudinger (1968) discovered a critical error in his proof. The combined work of Neil Trudinger, Thierry Aubin, and Richard Schoen later provided a complete solution to the problem.

The Yamabe problem is the following: given a smooth, compact manifold M of dimension n ≥ 3 with a Riemannian metric g, does there exist a metric g' conformal to g for which the scalar curvature of g' is constant? In other words, does a smooth function f exist on M for which the metric g' = e2fg has constant scalar curvature? The answer is now known to be yes, and was proved using techniques from differential geometry, functional analysis and partial differential equations.

The non-compact case

A closely related question is the so-called "non-compact Yamabe problem", which asks: on a smooth, complete Riemannian manifold (M,g) which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is no, due to counterexamples given by Jin (1988).

GPDE Workshop - Conformal invariants and nonlinear Yamabe problem on manifolds

Yamabe flow
Yamabe invariant

References

Lee, J.; Parker, T. (1987), "The Yamabe problem", Bulletin of the American Mathematical Society 17: 37–81, doi:10.1090/s0273-0979-1987-15514-5.
Trudinger, Neil S. (1968), "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds", Ann. Scuola Norm. Sup. Pisa (3) 22: 265–274, MR 0240748
Yamabe, Hidehiko (1960), "On a deformation of Riemannian structures on compact manifolds", Osaka Journal of Mathematics 12: 21–37, ISSN 0030-6126, MR 0125546
Schoen, Richard (1984), "Conformal deformation of a Riemannian metric to constant scalar curvature", J. Differential Geom. 20: 479–495.
Aubin, Thierry (1976), "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire", J. Math. Pures Appl., (9) 55: 269–296.