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Compactness results for complete metrics of constant positive scalar curvature on subdomains of \(S^ n\). (English) Zbl 0794.53025
We prove that the set of metrics conformal to the standard metric on \(S^ n\setminus \{p_ 1,\dots, p_ k\}\), where \(\{p_ 1,\dots, p_ k\}\) is an arbitrary set of \(k>1\) points, is compact in the \(C^{m,\alpha}\) topology provided the dilational Pokhozhaev invariants are bounded away from zero. Such metrics are necessarily complete on \(S^ n\setminus \{p_ 1,\dots, p_ k\}\) and asymptotic, at each singular points, to one of the periodic, spherically symmetric solutions on the positive \(n\)-dimensional half cylinder, which are known as Delaunay metrics. The Pokhozhaev invariants are defined at each singular point and their dilational component is shown to be equal to a dimensional constant times the Hamiltonian energy of the asymptotic Delaunay metric. We also address the issue of convergence when the invariants tend to zero for a proper subset of the singular points.

MSC:
53C20 Global Riemannian geometry, including pinching
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
58E11 Critical metrics
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