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' = 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.