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An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. (English) Zbl 0856.53028
This paper defines an invariant for certain vector fields called Yamabe-type flows, which have a conformal invariance built in, and uses it in order to give partial answers to scalar curvature problems in dimension $n\ge 7$. The scalar curvature of a metric conformal to the standard one on ${S}^{n}$ is assumed to have nondegenerate critical points of the variational problem. A Morse variational lemma at infinity is derived, which allows to see how the unstable manifold is built. The paper computes the intersection number of the critical points at infinity. The Yamabe flow becomes a pseudogradient for the variational problem, which satisfies the Palais-Smale condition on its decreasing flow lines.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58E05 Abstract critical point theory 53A55 Differential invariants (local theory), geometric objects 37C10 Vector fields, flows, ordinary differential equations