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
of dimension
, there exists a metric
conformal to
, such that the corresponding scalar curvature
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
and
on a manifold
are said to be
Note that, if
and
are conformally equivalent, then
is an isometry from
onto
. So we will only study below the case
.
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