Yamabe problem

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 $n\geq 3$ 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 there exist a smooth function f on M for which the metric $g'=e^{2f}g$ 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 well-known to be no, due to counterexamples given by Jin Zhiren.

See also

• Yamabe flow
• Yamabe invariant

References

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