Yamabe problem

The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. Yamabe (1960) claimed to have a solution, but Trudinger (1968) discovered a critical error in his proof. The combined work of Neil Trudinger, Thierry Aubin, and Richard Schoen later provided a complete solution to the problem in 1984.

The Yamabe problem is the following: Given a smooth, compact manifold M of dimension n ≥ 3 with a Riemannian metric g, does there exist a metric g' conformal to g for which the scalar curvature of g' is constant? In other words, does a smooth function f exist on M for which the metric g' = e^2fg has constant scalar curvature? The answer is now known to be yes, and was proved using techniques from differential geometry, functional analysis and partial differential equations.

Solutions of the Yamabe problem on a compact Einstein manifold

Let $(M,{\overline {g}})$ be a 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

${\overline {R}}_{ij}-{\frac {1}{n}}{\overline {R}}{\overline {g}}_{ij}=R_{ij}-{\frac {1}{n}}Rg_{ij}+{\frac {n-2}{\varphi }}{\Big (}\nabla _{i}\nabla _{j}\varphi +{\frac {1}{n}}g_{ij}\Delta \varphi {\Big )}.$

So, if ${\overline {\operatorname {Ric} }}={\frac {1}{n}}{\overline {R}}{\overline {g}}$ then

$\operatorname {Ric} -{\frac {1}{n}}Rg={\frac {2-n}{\varphi }}{\Big (}\operatorname {Hess} \varphi +{\frac {1}{n}}g\Delta \varphi {\Big )}$

and then

${\Big |}\operatorname {Ric} -{\frac {1}{n}}Rg{\Big |}^{2}={\frac {2-n}{\varphi }}{\Big (}\langle \operatorname {Ric} ,\operatorname {Hess} \varphi \rangle -{\frac {1}{n}}R\Delta \varphi {\Big )}.$

If $M$ is assumed compact, one can do an integration by parts, recalling the Bianchi identity $\operatorname {div} \operatorname {Ric} ={\frac {1}{2}}\nabla R,$ to see

$\int _{M}\varphi {\Big |}\operatorname {Ric} -{\frac {1}{n}}Rg{\Big |}^{2}\,d\mu _{g}={\frac {(n-2)^{2}}{2n}}\int _{M}\langle \nabla R,\nabla \varphi \rangle \,d\mu _{g}.$

This shows the following fact, due to Obata (1971):

• The only solutions to the Yamabe problem on a compact Einstein manifold are, themselves, Einstein.

Solutions of the Yamabe problem on a compact constant-curvature manifold

Let $(M,{\overline {g}})$ be a compact Riemannian manifold with constant curvature. Let $\varphi :M\to \mathbb {R}$ 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. According to the Weyl decomposition, then it has constant curvature. In summary:

• The only solutions to the Yamabe problem on a compact manifold with constant curvature have, themselves, 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. 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 $S^{n}.$ If $\varphi :S^{n}\to \mathbb {R}$ is a positive smooth function such that $\varphi ^{-2}{\overline {g}}$ has constant scalar curvature, there exists a positive number $\alpha$ and a diffeomorphism $\Phi :S^{n}\to S^{n}$ such that $\varphi ^{-2}{\overline {g}}=\alpha \Phi ^{\ast }{\overline {g}}.$

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.

See also

• Yamabe flow
• Yamabe invariant

References

• 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.
• 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
• 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
• Aubin, Thierry (1976), "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire", J. Math. Pures Appl., 55: 269–296
• 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
• 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