Yamabe problem

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The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:

Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature.

By computing a formula for how the scalar curvature of fg relates to that of g, this statement can be rephrased in the following form:

Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function φ on M, and a number c, such that

${\displaystyle {\frac {4(n-1)}{n-2}}\Delta ^{g}\varphi +R^{g}\varphi +c\varphi ^{(n+2)/(n-2)}=0.}$

Here n denotes the dimension of M, R^g denotes the scalar curvature of g, and ∆^g denotes the Laplace-Beltrami operator of g.

The mathematician Hidehiko Yamabe, in the paper Yamabe (1960), gave the above statements as theorems and provided a proof; however, Trudinger (1968) discovered an error in his proof. The problem of understanding whether the above statements are true or false became known as the Yamabe problem. The combined work of Yamabe, Trudinger, Thierry Aubin, and Richard Schoen provided an affirmative resolution to the problem in 1984.

It is now regarded as a classic problem in geometric analysis, with the proof requiring new methods in the fields of differential geometry and partial differential equations. A decisive point in Schoen's ultimate resolution of the problem was an application of the positive energy theorem of general relativity, which is a purely differential-geometric mathematical theorem first proved (in a provisional setting) in 1979 by Schoen and Shing-Tung Yau.

There has been more recent work due to Simon Brendle, Marcus Khuri, Fernando Codá Marques, and Schoen, dealing with the collection of all positive and smooth functions f such that, for a given Riemannian manifold (M,g), the metric fg has constant scalar curvature. Additionally, the Yamabe problem as posed in similar settings, such as for complete noncompact Riemannian manifolds, is not yet fully understood.