Yamabe problem


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:
By computing a formula for how the scalar curvature of relates to that of, this statement can be rephrased in the following form:
The mathematician Hidehiko Yamabe, in the paper, gave the above statements as theorems and provided a proof; however, 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 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 such that, for a given Riemannian manifold, the metric 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.

The Yamabe problem in special cases

Here, we refer to a "solution of the Yamabe problem" on a Riemmannian manifold as a Riemannian metric on for which there is a positive smooth function with

On a closed Einstein manifold

Let be a smooth Riemannian manifold. Consider a positive smooth function so that is an arbitrary element of the smooth conformal class of A standard computation shows
Taking the -inner product with results in
If is assumed to be Einstein, then the left-hand side vanishes. If is assumed to be closed, then one can do an integration by parts, recalling the Bianchi identity to see
If has constant scalar curvature, then the right-hand side vanishes. The consequent vanishing of the left-hand side proves the following fact, due to Obata :

On a closed constant-curvature manifold

Let be a closed Riemannian manifold with constant curvature. Let be a positive smooth function so that the Riemannian metric has constant scalar curvature. As established above, is an Einstein metric. Since it is conformal to a metric with vanishing Weyl curvature, it has vanishing Weyl curvature itself. By the Weyl decomposition, it follows that the assumptions of the Schur's lemma for the Riemann tensor are met; the conclusion of the Schur lemmma is that has constant curvature. In summary:
In the special case that is the standard -sphere, it follows that every solution to the Yamabe problem has constant positive curvature, since the -sphere does not support any metric of nonpositive curvature; otherwise there would be a contradiction to the Cartan-Hadamard theorem. Since every two Riemannian metrics on the sphere which have the same constant curvature are isometric, one can conclude:

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 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. Various additional criteria under which a solution to the Yamabe problem for a non-compact manifold can be shown to exist are known ; however, obtaining a full understanding of when the problem can be solved in the non-compact case remains a topic of research.

Research articles