The Yamabe problem on subdomains of even-dimensional spheres. (English) Zbl 0854.53037

Pacard, Frank

Topol. Methods Nonlinear Anal. 6, No. 1, 137-150 (1995).

We have seen some interesting works on the singular Yamabe problem. Here the author proves the existence of complete conformally flat metrics of constant positive scalar curvature on the complement in the standard sphere \(S^n\) of a finite number of \((n-2)/2\)-dimensional smooth submanifolds, provided \(n > 3\) is even. The proof is very similar to the proof of R. Mazzeo and N. Smale [J. Differ. Geom. 34, No. 3, 581-621 (1991; Zbl 0759.53029)]. This work generalizes the earlier result of the same author in dimensions four and six [Commun. Math. Phys. 159, No. 2, 423-432 (1994; Zbl 0822.35052)]. The new part here is the construction of the approximate solutions, relying on the work of P. Aviles before 1987.

Reviewer: Ma Li (Beijing)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

contraction; constant scalar curvature; singular Yamabe problem; approximate solutions

Citations:

Zbl 0759.53029; Zbl 0822.35052