Yamabe problem

Let be a compact Riemannian manifold of dimension ; let be its scalar curvature. The Yamabe problem is: Does there exist a metric , conformal to (cf. also Conformal-differential geometry), such that the scalar curvature of 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: must be replaced by , 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 (where , ), then the scalar curvature is given by

with . So, the Yamabe problem is equivalent to solving the equation

(a1)

Here, is the conformal Laplacian (cf. also Laplace operator). To solve this problem, Yamabe introduced the so-called Yamabe functional, , with .

The Euler equation of is (a1). Thus, the variational method seems applicable. Let be the infimum of for . One can prove that is a conformal invariant, and that is the infimum of over all , . But is the critical exponent in the Sobolev imbedding theorem (cf. Imbedding theorems), and is not compact. Hence one cannot prove that is attained. To overcome this difficulty, Yamabe considered the functionals , , and solved the family of approximated equations

(a2)

The following theorem holds, [a15]: There exists a strictly positive function , with and satisfying equation (a2), where for all , .

According to the sign of , there are three mutually exclusive cases: positive, negative and zero; has the sign of . Then Yamabe claimed that the set 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 , and satisfies (a1) with . In the positive case, if one considers the metric , then . 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 , one can exhibit subsequences with and such that satisfies weakly in . However, there are two difficulties: the regularity and the triviality of (according to the maximum principle, either or ). The regularity was resolved by N. Trudinger [a14]. He proved that a weak solution of (a1) is smooth. To prove that 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 , and such that is continuous but not compact and is compact. There are pairs of real numbers such that all satisfy . It can be proved that .

This situation occurs with , and . Moreover, Aubin proved that the best constant depends only on , and not upon the compact manifold. So, [a1], for any there exists a constant such that every satisfies .

Recently it has been proved [a8] that the best constant is achieved (i.e., exists). Using the above result, Aubin was able to prove the key theorem [a2]: satisfies (for , ). If , then there exists a strictly positive solution of with . For the metric one has . Here, is the volume of .

It remains to exhibit a test function such that . All subsequent work has centred on the discovery of appropriate test functions.

By considering the functional in a suitable conformal metric and using as test functions truncations of the functions (here, with a point at which the Weyl tensor is not zero), Aubin was able to prove [a2] that if () is a compact non-locally conformally flat Riemannian manifold, then . Hence there exists a conformal metric with .

The remaining cases.

In 1984, G. Medrano [a6] proved that for a large class of locally conformally-flat manifolds . At the same time, R. Schoen [a10] reduced the proof of to the proof of the positive mass conjecture. If is locally conformally flat, there is a conformal metric such that is flat in a neighbourhood of . In the positive case the conformal Laplacian is invertible; let be its inverse. The expansion of in is

(a3)

where and is a harmonic function.

When , in a conformal metric , has an expansion like (a3); when , has the form (a3); is or Lipschitzian according to the dimension. In [a10] it is proved that if , then ; it uses test functions equal to in a neighbourhood of and equal to when is large. It remained to proved that if is not conformal to . This was done for in [a12]. For locally conformally-flat manifolds the result is in [a13], and for dimensions smaller than in [a11]. For a unification of the work of Aubin and Schoen, see [a9].

There are also direct proofs, not considering the functions ; one proceeds by successive approximation, the other by the blow-up method. These proofs use the value of the best constant . 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 is achieved.

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

Let be a compact Riemannian manifold of dimension which is not conformal to . There exists a conformal metric with for which any conformal transformation is an isometric mapping .

A generalization of the Yamabe problem is the prescribed scalar curvature problem in a given conformal class. This problem on 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 .

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