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*' = *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: 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).