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 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 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.
Here, we refer to a "solution of the Yamabe problem" on a Riemannian manifold
(M,\overline{g})
\varphi:M\toR,
g=\varphi-2\overline{g}.
Let
(M,\overline{g})
\varphi:M\toR,
g=\varphi-2\overline{g}
\overline{g}.
\overline{R}ij-
| 1 | |
| n |
\overline{R}\overline{g}ij=Rij-
| 1 | |
| n |
Rgij+
| n-2 | |
| \varphi |
(\nablai\nabla
|
gij\Delta\varphi).
|
| \varphi\left\langle\overline{\operatorname{Ric}}- | 1 | \overline{R}\overline{g},\operatorname{Ric}- |
| n |
| 1 | |
| n |
| Rg\right\rangle | ||||
|
| 2+(n-2)(\langle\operatorname{Ric},\operatorname{Hess}\varphi\rangle | ||||
| Rg| | ||||
|
R\Delta\varphi).
\overline{g}
M
|
R,
\intM\varphi|\operatorname{Ric}-
| 1 | |
| n |
| 2d\mu | |||||
| Rg| | - | ||||
|
| 1 | |
| n |
)\intM\langle\nablaR,\nabla\varphi\rangled\mug.
Obata then went on to prove that, except in the case of the standard sphere with its usual constant-sectional-curvature metric, the only constant-scalar-curvature metrics in the conformal class of an Einstein metric (on a closed manifold) are constant multiples of the given metric. The proof proceeds by showing that the gradient of the conformal factor is actually a conformal Killing field. If the conformal factor is not constant, following flow lines of this gradient field, starting at a minimum of the conformal factor, then allows one to show that the manifold is conformally related to the cylinder
Sn-1 x R
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 (for example); however, obtaining a full understanding of when the problem can be solved in the non-compact case remains a topic of research.