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:

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

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

The Yamabe problem in special cases

Here, we refer to a "solution of the Yamabe problem" on a Riemmannian manifold $(M,{\overline {g}})$ as a Riemannian metric $g$ on $M$ for which there is a positive smooth function $\varphi :M\to \mathbb {R} ,$ with $g=\varphi ^{-2}{\overline {g}}.$

On a closed Einstein manifold

Let $(M,{\overline {g}})$ be a smooth Riemannian manifold. Consider a positive smooth function $\varphi :M\to \mathbb {R} ,$ so that $g=\varphi ^{-2}{\overline {g}}$ is an arbitrary element of the smooth conformal class of ${\overline {g}}.$ A standard computation shows

{\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 )}.

Taking the $g$ -inner product with $\textstyle \varphi (\operatorname {Ric} -{\frac {1}{n}}Rg)$ results in

\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 )}.

If ${\overline {g}}$ is assumed to be Einstein, then the left-hand side vanishes. If $M$ is assumed to be closed, then one can do an integration by parts, recalling the Bianchi identity $\textstyle \operatorname {div} \operatorname {Ric} ={\frac {1}{2}}\nabla R,$ to see

\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}.

If $R$ 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):

Every solution to the Yamabe problem on a closed Einstein manifold is Einstein.

On a closed constant-curvature manifold

Let $(M,{\overline {g}})$ be a closed Riemannian manifold with constant curvature. Let $\textstyle \varphi :M\to \mathbb {R}$ be a positive smooth function so that the Riemannian metric $\varphi ^{-2}{\overline {g}}$ has constant scalar curvature. As established above, $\varphi ^{-2}{\overline {g}}$ 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 $\varphi ^{-2}{\overline {g}}$ has constant curvature. In summary:

Every solution to the Yamabe problem on a closed manifold with constant curvature has constant curvature.

In the special case that $(M,{\overline {g}})$ is the standard $n$ -sphere, it follows that every solution to the Yamabe problem has constant positive curvature, since the $n$ -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:

Let ${\overline {g}}$ denote the standard Riemannian metric on $S^{n}.$ Every solution to the Yamabe problem on $(S^{n},{\overline {g}})$ is of the form $\alpha \Phi ^{\ast }{\overline {g}}$ for a positive number $\alpha$ and a diffeomorphism $\Phi :S^{n}\to S^{n}$ .

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 $(M, g)$ 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 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 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.

References

Research articles

Aubin, Thierry (1976), "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire", J. Math. Pures Appl., 55: 269–296
Aviles, P.; McOwen, R. C. (1988), "Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds", J. Differ. Geom., 27 (2): 225–239, doi:10.4310/jdg/1214441781, MR 0925121
Jin, Zhiren (1988), "A counterexample to the Yamabe problem for complete noncompact manifolds", Lect. Notes Math., Lecture Notes in Mathematics, 1306: 93–101, doi:10.1007/BFb0082927, ISBN 978-3-540-19097-4
Lee, John M.; Parker, Thomas H. (1987), "The Yamabe problem", Bulletin of the American Mathematical Society, 17: 37–81, doi:10.1090/s0273-0979-1987-15514-5.
Obata, Morio (1971), "The conjectures on conformal transformations of Riemannian manifolds", J. Differential Geometry, 6: 247–258, doi:10.4310/jdg/1214430407, MR 0303464
Schoen, Richard (1984), "Conformal deformation of a Riemannian metric to constant scalar curvature", J. Differ. Geom., 20 (2): 479–495, doi:10.4310/jdg/1214439291
Trudinger, Neil S. (1968), "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds", Ann. Scuola Norm. Sup. Pisa (3), 22: 265–274, MR 0240748
Yamabe, Hidehiko (1960), "On a deformation of Riemannian structures on compact manifolds", Osaka Journal of Mathematics, 12: 21–37, ISSN 0030-6126, MR 0125546

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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