Yamabe problem

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The Yamabe problem in differential geometry takes its name from the mathematician Hidehiko Yamabe. Although Yamabe claimed to have a solution in 1960, a critical error in his proof was discovered in 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 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 there exist a smooth function f 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.

[edit] 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 well-known to be no, due to counterexamples given by Jin Zhiren.

[edit] See also

• Yamabe flow
• Yamabe invariant

[edit] References

• J. Lee and T. Parker, "The Yamabe problem", Bull. Amer. Math. Soc. 17, 37-81 (1987).