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. Although Yamabe (1960) claimed to have a solution in 1960, a critical error in his proof was discovered by Trudinger (1968). The combined work of Neil Trudinger, Thierry Aubin, and Richard Schoen provided a complete solution to the problem as of 1984.

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)

See also

• Yamabe flow
• Yamabe invariant

References

• Lee, J.; Parker, T. (1987), "The Yamabe problem", Bulletin of the American Mathematical Society 17: 37–81, http://www.ams.org/bull/1987-17-01/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, MR0240748, http://www.numdam.org/item?id=ASNSP_1968_3_22_2_265_0 
• Yamabe, Hidehiko (1960), "On a deformation of Riemannian structures on compact manifolds", Osaka Journal of Mathematics 12: 21–37, ISSN 0030-6126, MR0125546, http://projecteuclid.org/euclid.ojm/1200689814 
• Jin, Zhiren (1988), "A counterexample to the Yamabe problem for complete noncompact manifolds", Partial differential equations (Tianjin, 1986), Lecture Notes in Math., 1306, Berlin, New York: Springer-Verlag, pp. 93–101, doi:10.1007/BFb0082927, MR1032773