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 \(n\geq 7\). 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.
Reviewer: S.-D.Cartianu (Bucureşti)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 53A55 | Differential invariants (local theory), geometric objects |
| 37C10 | Dynamics induced by flows and semiflows |
Keywords:
Yamabe-type flows; scalar curvature problems; intersection number; critical points at infinityReferences:
| [1] | A. Bahri, The scalar-curvature problem on spheres of dimension \(n\geqslant7\) , · Zbl 0804.53053 |
| [2] | A. Bahri, Letter to various mathematicians pointing out a gap in [1]. · Zbl 1193.42069 |
| [3] | A. Bahri, Critical points at infinity in some variational problems , Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow, 1989. · Zbl 0676.58021 |
| [4] | A. Bahri and P. H. Rabinowitz, Periodic solutions of Hamiltonian systems of \(3\)-body type , Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 6, 561-649. · Zbl 0745.34034 |
| [5] | J. Milnor, Lectures on the \(h\)-cobordism theorem , Notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J., 1965. · Zbl 0161.20302 |
| [6] | A. Floer, Cuplength estimates on Lagrangian intersections , Comm. Pure Appl. Math. 42 (1989), no. 4, 335-356. · Zbl 0683.58017 · doi:10.1002/cpa.3160420402 |
| [7] | C. H. Taubes, Path connected Yang-Mills moduli spaces , J. Differential Geom. 19 (1984), no. 2, 337-392. · Zbl 0551.53040 |
| [8] | 1 P. L. Lions, The concentration compactness principle in the calculus of variations. The limit case. I , Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201. · Zbl 0704.49005 · doi:10.4171/RMI/6 |
| [9] | 2 P. L. Lions, The concentration compactness principle in the calculus of variations. The limit case. II , Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121. · Zbl 0704.49006 · doi:10.4171/RMI/12 |
| [10] | A. Bahri, Addenda to the book Critical Points at Infinity in Some Variational Problems [3] and to the paper “The scalar-curvature problem on the standard three-dimensional sphere” , · Zbl 1184.53083 |
| [11] | A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain , Comm. Pure Appl. Math. 41 (1988), no. 3, 253-294. · Zbl 0649.35033 · doi:10.1002/cpa.3160410302 |
| [12] | A. Bahri and H. Brezis, Nonlinear elliptic equations involving the critical Sobolev exponent on manifolds , C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 537-576. Related paper to appear in a volume dedicated to the memory of E. D’Atri. · Zbl 0694.35059 |
| [13] | A. Bahri and P. L. Lions, On the existence of a positive solution to semi-linear elliptic equations in \(\mathbbR^n\) , · Zbl 0883.35045 |
| [14] | M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities , Math. Z. 187 (1984), no. 4, 511-517. · doi:10.1007/BF01174186 |
| [15] | A. Bahri, Y. Chen, and L. Ma, Multiplicity results for the scalar-curvature problem in dimensions \(3\) and \(4\) , |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
