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 in 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' = 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: Is it true that on every smooth complete Riemannian manifold (M,g) which is not compact, there exists a metric that is conformal to g, has constant scalar curvature and is also complete? The answer is no, due to counterexamples given by Jin (1988).