Ngô Quốc Anh

June 28, 2011

The Yamabe problem: A Story

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:47

I want to write a short survey about the Yamabe problem. Long time ago, I introduced the problem in this blog [here] but it turns out that the note was not rich enough to perform the importance of the problem.

Hidehiko Yamabe, in his famous paper entitled On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), pp. 21-37,  wanted to solve the Poincaré conjecture

Conjecture. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

For this he thought, as a first step, to exhibit a metric with constant scalar curvature. We refer the reader to this note for details. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement:

Theorem (Yamabe). On a compact Riemannian manifold (M, g) of dimension \geqslant 3, there exists a metric g' conformal to g, such that the corresponding scalar curvature \text{Scal}_{g'} is constant.

As can be seen, the Yamabe problem is a special case of the prescribing scalar curvature problem that can be completely solved. For the prescribing scalar curvature, we also solve it completely when the invariant is non-positive.

1. Conformal metrics.

Definition (conformal). Two pseudo-Riemannian metrics g and \widetilde g on a manifold M are said to be

  • (pointwise) conformal if there exists a C^\infty function f on M such that

    \displaystyle \widetilde g=e^{2f}g;

  • conformally equivalent if there exists a diffeomorphism \alpha of M such that \alpha^* \widetilde g and g are pointwise conformal.

Note that, if g and \widetilde g are conformally equivalent, then \alpha is an isometry from e^{2f}g onto \widetilde g. So we will only study below the case \widetilde g = e^{2f}g.


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