Yamabe problem

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' = e^2fg 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

See also

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.