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Prescribed scalar curvature on the \(n\)-sphere. (English) Zbl 0843.53037

The authors establish the Morse inequalities for the scalar curvature problem (i.e. The Kazdan-Warner problem) on \(S^3\). This result compliments an earlier existence theorem of A. Bahri and J. M. Coron [J. Funct. Anal. 95, No. 1, 106-172 (1991; Zbl 0722.53032)]. Because of the failure of the Palais-Smale condition for the corresponding variational problem, they study the behavior of critical points of functionals \((K_p)\) with subcritical exponents \((p)\) go to the critical exponent. The key results in getting Morse theory are contained in their Theorem 2.4 and Theorem 3.2. To obtain these results, they give a very delicate analysis. This kind of analysis was also provided by Yanyan Li [J. Differ. Equations 120, No. 2, 319-410 (1995; Zbl 0827.53039)]. Their Morse inequalities can read as follows. Suppose \(K\) is a given positive function on \(S^3\) and it is also a regular value of the scalar curvature map. Let \(D_\mu\) be the number of critical points of \(K\) on \(S^3\) at which the Laplacian of \(K\) is negative and for which the Morse index of \(-K\) is \(\mu\). Then there holds: \[ (-1)^\lambda\leq \sum^\lambda_{\mu=0} (-1)^{\lambda- \mu} (C_\mu D_\mu), \qquad \lambda= 0, 1, 2, \dots \] and \[ 1= \sum^\infty_{\mu=0} (-1)^\mu C_\mu \sum^2_{\mu=0} (-1)^\mu D_\mu, \] where \(C_\mu\) is the number of solutions of the curvature \(K\) with Morse index \(C_\mu\) \((C_\mu=0\) for \(\mu\) large). We remark here that these Morse inequalities can also be obtained by the original method of A. Bahri and J. M. Coron cited above. This paper will be of interest to many researchers in this field.
Reviewer: Ma Li (Beijing)

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E11 Critical metrics
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