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{{More footnotes needed|date=September 2018}}
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 manifold]]s:
{{blockquote|Let {{math|(''M'',''g'')}} be a closed smooth Riemannian manifold. Then there exists a positive and smooth function {{mvar|f}} on {{mvar|M}} such that the Riemannian metric {{math|''fg''}} has constant scalar curvature.}}
By computing a formula for how the scalar curvature of {{math|''fg''}} relates to that of {{mvar|g}}, this statement can be rephrased in the following form:
{{blockquote|Let {{math|(''M'',''g'')}} be a closed smooth Riemannian manifold. Then there exists a positive and smooth function {{mvar|φ}} on {{mvar|M}}, and a number {{mvar|c}}, such that
: <math>\frac{4(n-1)}{n-2}\Delta^g\varphi+R^g\varphi+c\varphi^{(n+2)/(n-2)}=0.</math>
Here {{mvar|n}} denotes the dimension of {{mvar|M}}, {{math|''R''<sup>''g''</sup>}} denotes the scalar curvature of {{mvar|g}}, and {{math|∆<sup>''g''</sup>}} denotes the Laplace-Beltrami operator of {{mvar|g}}.}}

The mathematician [[Hidehiko Yamabe]], in the paper {{harvtxt|Yamabe|1960}}, gave the above statements as theorems and provided a proof; however, {{harvtxt|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 {{mvar|f}} such that, for a given Riemannian manifold {{math|(''M'',''g'')}}, the metric {{math|''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.

==The Yamabe problem in special cases==
Here, we refer to a "solution of the Yamabe problem" on a Riemannian manifold <math>(M,\overline{g})</math> as a Riemannian metric {{mvar|g}} on {{mvar|M}} for which there is a positive smooth function <math>\varphi:M\to\mathbb{R},</math> with <math>g=\varphi^{-2}\overline{g}.</math>

===On a closed Einstein manifold===
Let <math>(M,\overline{g})</math> be a smooth Riemannian manifold. Consider a positive smooth function <math>\varphi:M\to\mathbb{R},</math> so that <math>g=\varphi^{-2}\overline{g}</math> is an arbitrary element of the smooth conformal class of <math>\overline{g}.</math> A standard computation shows
: <math>\overline{R}_{ij}-\frac{1}{n}\overline{R}\overline{g}_{ij}=R_{ij}-\frac{1}{n}Rg_{ij}+\frac{n-2}{\varphi}\Big(\nabla_i\nabla_j\varphi+\frac{1}{n}g_{ij}\Delta\varphi\Big).</math>
Taking the {{mvar|g}}-inner product with <math>\textstyle\varphi(\operatorname{Ric}-\frac{1}{n}Rg)</math> results in
:<math>\varphi\left\langle\overline{\operatorname{Ric}}-\frac{1}{n}\overline{R}\overline{g},\operatorname{Ric}-\frac{1}{n}Rg\right\rangle_g=\varphi\Big|\operatorname{Ric}-\frac{1}{n}Rg\Big|_g^2+(n-2)\Big(\big\langle\operatorname{Ric},\operatorname{Hess}\varphi\big\rangle_g-\frac{1}{n}R\Delta\varphi\Big).</math>
If <math>\overline{g}</math> is assumed to be Einstein, then the left-hand side vanishes. If <math>M</math> is assumed to be closed, then one can do an [[integration by parts]], recalling the Bianchi identity <math>\textstyle\operatorname{div}\operatorname{Ric}=\frac{1}{2}\nabla R,</math> to see
: <math>\int_M \varphi\Big|\operatorname{Ric}-\frac{1}{n}Rg\Big|^2\,d\mu_g=(n-2)\Big(\frac{1}{2}-\frac{1}{n}\Big)\int_M \langle\nabla R,\nabla\varphi\rangle\,d\mu_g.</math>
If {{mvar|g}} 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 (1971):
{{blockquote|Every solution to the Yamabe problem on a closed Einstein manifold is Einstein.}}

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 <math>S^{n-1}\times \mathbb{R}</math>, and hence has vanishing Weyl curvature.

==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]] {{math|(''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 {{harvtxt|Jin|1988}}. 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 {{harvtxt|Aviles|McOwen|1988}}); however, obtaining a full understanding of when the problem can be solved in the non-compact case remains a topic of research.

==See also==
* [[Yamabe flow]]
* [[Yamabe invariant]]

==References==
===Research articles===
*{{Citation | last1=Aubin | first1=Thierry | title= Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire | year=1976| journal=J. Math. Pures Appl.|volume=55 |pages=269–296}} 
*{{citation| last1=Aviles| first1 = P. | last2=McOwen| first2 = R. C.|title =  Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds| journal= J. Differ. Geom. |volume=27 | issue = 2 |pages=225–239|year=1988|mr=0925121| doi = 10.4310/jdg/1214441781 | doi-access=free}}
*{{citation
 | last = Jin | first = Zhi Ren
 | contribution = A counterexample to the Yamabe problem for complete noncompact manifolds
 | doi = 10.1007/BFb0082927
 | mr = 1032773
 | pages = 93–101
 | publisher = Springer | location = Berlin
 | series = Lecture Notes in Mathematics
 | title = Partial differential equations (Tianjin, 1986)
 | volume = 1306
 | isbn = 978-3-540-19097-4
 | year = 1988}}
*{{citation|first1=John M.|last1=Lee|first2=Thomas H.|last2=Parker|url=http://www.ams.org/bull/1987-17-01/S0273-0979-1987-15514-5/|title=The Yamabe problem|journal=Bulletin of the American Mathematical Society|volume=17|pages=37–81|year=1987|doi=10.1090/s0273-0979-1987-15514-5|doi-access=free}}.
*{{citation|first1=Morio|last1=Obata|title=The conjectures on conformal transformations of Riemannian manifolds|journal=Journal of Differential Geometry|volume=6|pages=247–258|year=1971|issue=2 |doi=10.4310/jdg/1214430407|mr=0303464|doi-access=free}}
*{{Citation | last1=Schoen  | first1=Richard  | title=Conformal deformation of a Riemannian metric to constant scalar curvature |year=1984| journal=J. Differ. Geom. |volume= 20 | issue=2 | pages= 479–495| doi=10.4310/jdg/1214439291 | doi-access=free }}
*{{Citation | last1=Trudinger | first1=Neil S. | author1-link=Neil Trudinger | title=Remarks concerning the conformal deformation of Riemannian structures on compact manifolds | url=http://www.numdam.org/item?id=ASNSP_1968_3_22_2_265_0 | mr=0240748 | year=1968 | journal=Ann. Scuola Norm. Sup. Pisa (3) | volume=22 | pages=265–274}}
*{{Citation | last1=Yamabe | first1=Hidehiko | title=On a deformation of Riemannian structures on compact manifolds | url=http://projecteuclid.org/euclid.ojm/1200689814 | mr=0125546 | year=1960 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=12 | pages=21–37}}

===Textbooks===
* Aubin, Thierry. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xviii+395 pp. {{ISBN|3-540-60752-8}}
* Schoen, R.; Yau, S.-T. Lectures on differential geometry. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. With a preface translated from the Chinese by Kaising Tso. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. v+235 pp. {{ISBN|1-57146-012-8}}
* Struwe, Michael. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 34. Springer-Verlag, Berlin, 2008. xx+302 pp. {{ISBN|978-3-540-74012-4}}

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[[Category:Riemannian geometry]]
[[Category:Mathematical problems]]