Yamabe problem

Let $ ( M _ {n} ,g ) $ be a $ C ^ \infty $ compact Riemannian manifold of dimension $ n \geq 3 $; let $ R $ be its scalar curvature. The Yamabe problem is: Does there exist a metric $ g ^ \prime $, conformal to $ g $( cf. also Conformal-differential geometry), such that the scalar curvature $ R ^ \prime $ of $ g ^ \prime $ is constant?

In 1960, H. Yamabe wanted to solve the Poincaré conjecture. As a first step he tried to make constant the scalar curvature by a conformal change of metrics. He thought he had succeeded. Unfortunately, in his beautiful paper [a15] there is a mistake in an inequality: $ \| {v ^ {( q ) } } \| _ {q _ {n} } \leq \textrm{ const } \| {v ^ {( q ) } } \| _ {q _ {1} } $ must be replaced by $ \| {v ^ {( q ) } } \| _ {q _ {n} } \leq \textrm{ const } \| {v ^ {( q ) } } \| _ {q _ {1} } ^ {( q - 1 ) ^ {n - 1 } } $, and this does not yield the result in the general case. Now, thirty years afterwards, the problem is entirely solved.

Yamabe was a pioneer of solving geometrical problems by analysis. If one writes the conformal deformation in the form $ g ^ \prime = \phi ^ { {4 / {( n - 2 ) } } } g $( where $ \phi \in C ^ \infty $, $ \phi > 0 $), then the scalar curvature $ R ^ \prime $ is given by

$$ R ^ \prime = \phi ^ {- { {( n + 2 ) } / {( n + 2 ) } } } \left [ R \phi + { \frac{4 ( n - 1 ) \Delta \phi }{n - 2 } } \right ] $$

with $ \Delta \phi = - \nabla ^ {i} \nabla _ {i} \phi $. So, the Yamabe problem is equivalent to solving the equation

$$ \tag{a1 } L \phi = \epsilon \phi ^ { {{( n + 2 ) } / {( n - 2 ) } } } , \phi > 0, \textrm{ with } \epsilon = 1,0, - 1. $$

Here, $ L = \Delta \phi + { {( n - 2 ) R } / {4 ( n - 1 ) } } $ is the conformal Laplacian (cf. also Laplace operator). To solve this problem, Yamabe introduced the so-called Yamabe functional, $ J ( \phi ) = \| \phi \| _ {N} ^ {- 2 } \int _ {V} {\phi L \phi } {d V } $, with $ N = { {2n } / {( n - 2 ) } } $.

The Euler equation of $ J ( \phi ) $ is (a1). Thus, the variational method seems applicable. Let $ \mu $ be the infimum of $ J ( \phi ) $ for $ \phi \in {\mathcal A} = \{ {\psi \in H _ {1} } : {\psi \geq 0, \psi \not\equiv 0 } \} $. One can prove that $ \mu $ is a conformal invariant, and that $ \mu $ is the infimum of $ J ( \phi ) $ over all $ \phi \in H _ {1} $, $ \phi \not\equiv 0 $. But $ N $ is the critical exponent in the Sobolev imbedding theorem (cf. Imbedding theorems), and $ H _ {1} \subset L _ {N} $ is not compact. Hence one cannot prove that $ \mu $ is attained. To overcome this difficulty, Yamabe considered the functionals $ J _ {q} ( \psi ) = \| \psi \| _ {q} ^ {- 2 } \int _ {V} {\psi L \psi } {d V } $, $ 2 < q < N $, and solved the family of approximated equations

$$ \tag{a2 } L \phi = \mu _ {q} \phi ^ {q - 1 } , \phi > 0, 2 < q < N. $$

The following theorem holds, [a15]: There exists a $ C ^ \infty $ strictly positive function $ \phi _ {q} $, with $ \| {\phi _ {q} } \| _ {q} = 1 $ and satisfying equation (a2), where $ \mu _ {q} = J _ {q} ( \phi _ {q} ) = \inf J _ {q} ( \psi ) $ for all $ \psi \in H _ {1} $, $ \psi \not\equiv 0 $.

According to the sign of $ \mu $, there are three mutually exclusive cases: positive, negative and zero; $ \mu _ {q} $ has the sign of $ \mu $. Then Yamabe claimed that the set $ \{ \phi _ {q} \} $ is uniformly bounded. This is not true on the sphere, and this cannot be overcome in the positive case. But in the negative case the wrong term plays no role, and one can remove it (it has negative sign). Yamabe's proof works also in the zero case: if $ \mu _ {q} = 0 $, $ L \phi _ {q} = 0 $ and $ \phi _ {q} $ satisfies (a1) with $ \epsilon = 0 $. In the positive case, if one considers the metric $ {\widetilde{g} } = \phi _ {q} ^ { {4 / {( n - 2 ) } } } g $, then $ {\widetilde{R} } = \mu _ {q} \phi _ {q} ^ {q - N } > 0 $. So Yamabe was able to prove that there exists a conformal metric whose scalar curvature is either a non-positive constant or is everywhere positive.

The positive case remained open. When $ \mu > 0 $, one can exhibit subsequences $ \{ \phi _ {q _ {i} } \} $ with $ q _ {i} \rightarrow N $ and $ \psi \in H _ {1} $ such that $ \psi \geq 0 $ satisfies $ L \psi = \psi ^ {N - 1 } $ weakly in $ H _ {1} $. However, there are two difficulties: the regularity and the triviality of $ \psi $( according to the maximum principle, either $ \psi > 0 $ or $ \psi \equiv 0 $). The regularity was resolved by N. Trudinger [a14]. He proved that a weak solution of (a1) is smooth. To prove that $ \psi $ is non-trivial, the best constants in the Sobolev imbedding theorem must be found. In [a1] (see also [a3]), T. Aubin considered three Banach spaces $ B _ {1} $, $ B _ {2} $ and $ B _ {3} $ such that $ B _ {1} \subset B _ {2} $ is continuous but not compact and $ B _ {1} \subset B _ {3} $ is compact. There are pairs of real numbers $ ( C,A ) $ such that all $ x \in B _ {1} $ satisfy $ \| x \| _ {B _ {2} } \leq C \| x \| _ {B _ {1} } + A \| x \| _ {B _ {3} } $. It can be proved that $ K = \inf \{ C : {\exists A } \} > 0 $.

This situation occurs with $ H _ {1} $, $ L _ {N} $ and $ L _ {2} $. Moreover, Aubin proved that the best constant $ K ( n,2 ) $ depends only on $ n $, and not upon the compact manifold. So, [a1], for any $ \epsilon > 0 $ there exists a constant $ A ( \epsilon ) $ such that every $ \phi \in H _ {1} $ satisfies $ \| \phi \| _ {N} \leq [ K ( n,2 ) + \epsilon ] \| {\nabla \phi } \| _ {2} + A ( \epsilon ) \| \phi \| _ {2} $.

Recently it has been proved [a8] that the best constant is achieved (i.e., $ A ( 0 ) $ exists). Using the above result, Aubin was able to prove the key theorem [a2]: $ \mu $ satisfies $ \mu \leq \mu _ {S} = { {n ( n - 2 ) \omega _ {n} ^ { {2 / n } } } / 4 } $( for $ ( S _ {n} , \textrm{ can } ) $, $ \mu = \mu _ {S} $). If $ \mu < \mu _ {S} $, then there exists a strictly positive $ C ^ \infty $ solution $ \psi $ of $ L \psi = \mu \psi ^ {N - 1 } $ with $ \| \psi \| _ {N} = 1 $. For the metric $ {\widetilde{g} } = \psi ^ { {4 / {( n - 2 ) } } } g $ one has $ {\widetilde{R} } = \mu $. Here, $ \omega _ {n} $ is the volume of $ S _ {n} ( 1 ) $.

It remains to exhibit a test function $ \psi $ such that $ J ( \psi ) < \mu _ {S} $. All subsequent work has centred on the discovery of appropriate test functions.

By considering the functional $ J $ in a suitable conformal metric and using as test functions truncations of the functions $ ( r ^ {2} + \epsilon ) ^ {1 - {n / 2 } } $( here, $ r ( Q ) = d ( P,Q ) $ with $ P $ a point at which the Weyl tensor is not zero), Aubin was able to prove [a2] that if $ ( M _ {n} ,g ) $( $ n \geq 6 $) is a compact non-locally conformally flat Riemannian manifold, then $ \mu < \mu _ {S} $. Hence there exists a conformal metric $ g ^ \prime $ with $ R ^ \prime = \mu $.

The remaining cases.

In 1984, G. Medrano [a6] proved that for a large class of locally conformally-flat manifolds $ \mu < \mu _ {S} $. At the same time, R. Schoen [a10] reduced the proof of $ \mu < \mu _ {S} $ to the proof of the positive mass conjecture. If $ ( M _ {n} ,g ) $ is locally conformally flat, there is a conformal metric $ {\widetilde{g} } $ such that $ {\widetilde{g} } $ is flat in a neighbourhood $ \theta $ of $ P $. In the positive case the conformal Laplacian is invertible; let $ G $ be its inverse. The expansion of $ G ( x ) $ in $ \theta $ is

$$ \tag{a3 } G ( x ) = { \frac{r ^ {2 - n } + \alpha ( x ) }{n - 2 } } \omega _ {n - 1 } , $$

where $ r = d ( P,x ) $ and $ \alpha ( x ) $ is a harmonic function.

When $ n = 4, 5 $, in a conformal metric $ {\widetilde{g} } $, $ G _ { {\widetilde{L} } } $ has an expansion like (a3); when $ n = 3 $, $ G _ {L} $ has the form (a3); $ \alpha ( x ) $ is $ C ^ {1} $ or Lipschitzian according to the dimension. In [a10] it is proved that if $ A = \alpha ( P ) > 0 $, then $ \mu < \mu _ {S} $; it uses test functions equal to $ ( \epsilon + r ^ {2} ) ^ {1 - {n / 2 } } $ in a neighbourhood of $ P $ and equal to $ \epsilon _ {0} G ( x ) $ when $ r $ is large. It remained to proved that $ A > 0 $ if $ ( M _ {n} ,g ) $ is not conformal to $ ( S _ {n} , \textrm{ can } ) $. This was done for $ n = 3 $ in [a12]. For locally conformally-flat manifolds the result is in [a13], and for dimensions smaller than $ 7 $ in [a11]. For a unification of the work of Aubin and Schoen, see [a9].

There are also direct proofs, not considering the functions $ \phi _ {q} $; one proceeds by successive approximation, the other by the blow-up method. These proofs use the value of the best constant $ K _ {n,2 } $. In [a4], A. Bahri presents an algebraic-topological proof for locally conformally-flat manifolds, not using the positive mass conjecture. Here it can not be shown that $ \mu $ is achieved.

In [a7], both the Yamabe problem and the Lichnerowicz problem are solved.

Let $ ( M _ {n} ,g ) $ be a compact Riemannian manifold of dimension $ n \geq 3 $ which is not conformal to $ ( S _ {n} , \textrm{ can } ) $. There exists a conformal metric $ {\widetilde{g} } $ with $ {\widetilde{R} } = \textrm{ const } $ for which any conformal transformation is an isometric mapping $ ( I, ( M, {\widetilde{g} } ) = C ( M,g ) ) $.

A generalization of the Yamabe problem is the prescribed scalar curvature problem in a given conformal class. This problem on $ S _ {n} $ is known as the Nirenberg problem. Although research is intensive, these problems have not yet (1996) been entirely solved. On compact manifolds with boundary, P. Cherrier presented an original problem in [a5]: To find a conformal metric with prescribed scalar curvature and prescribed mean curvature of $ \partial M $.

References

[a1]	T. Aubin, "Espaces de Sobolev sur les variétés Riemanniennes" Bull. Sci. Math. , 100 (1976) pp. 149–173
[a2]	T. Aubin, "Équations différentielles non-linéaires et problème de Yamabe concernant la courbure scalaire" J. Math. Pures Appl. , 55 (1976) pp. 269–296
[a3]	T. Aubin, "Nonlinear analysis on manifolds. Monge–Ampère equations" , Springer (1982)
[a4]	A. Bahri, "Proof of the Yamabe conjecture, without the positive mass theorem, for locally conformally flat manifolds" T. Mabuchi (ed.) S. Mukai (ed.) , Einstein Metrics and Yang–Mills Connections , M. Dekker (1993)
[a5]	P. Cherrier, "Problèmes de Neumann non-linéaires sur les variétés Riemanniennes" J. Funct. Anal. , 57 (1984) pp. 154–206
[a6]	O.G. Medrano, "On the Yamabe problem concerning the compact locally conformally flat manifolds" J. Funct. Anal. , 66 (1986) pp. 42–53
[a7]	E. Hebey, M Vaugon, "Le problème de Yamabe équivariant" Bull. Sci. Math. , 117 (1993) pp. 241–286
[a8]	E. Hebey, M. Vaugon, "The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds" Duke Math. J. , 79 (1995) pp. 235–279
[a9]	J.M. Lee, T.H. Parker, "The Yamabe problem" Bull. Amer. Math. Soc. , 17 (1987) pp. 37–91
[a10]	R. Schoen, "Conformal deformation of a Riemannian metric to constant scalar curvature" J. Diff. Geom. , 20 (1984) pp. 479–495
[a11]	R. Schoen, "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics" , Topics in Calculus of Variations , Lecture Notes in Mathematics , 1365 , Springer (1989)
[a12]	R. Schoen, S.T. Yau, "On the proof of the positive mass conjecture in general relativity" Comm. Math. Phys. , 65 (1979) pp. 45–76
[a13]	R. Schoen, S.T. Yau, "Conformally flat manifolds, Kleinian groups and scalar curvature" Invent. Math. , 92 (1988) pp. 47–71
[a14]	N. Trudinger, "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds" Ann. Scuola Norm. Sup. Pisa , 22 (1968) pp. 265–274
[a15]	H. Yamabe, "On a deformation of Riemannian strctures on compact manifolds" Osaka Math. J. , 12 (1960) pp. 21–37