Isolation of the Weyl conformal tensor for Einstein manifolds. (Q1869654)

The authors give a generalization of the isolation theorem of \textit{M. A. Singer} [ Differ. Geom. Appl. 2, 269--274 (1992; Zbl 0741.53035)]. More exactly, they prove that the Weyl conformal tensor of an oriented positive Ricci Einstein \(n\)-manifold (\(n\geq 4\)) obeys the following isolation theorem. Theorem. Let \((M,g)\) be a compact connected oriented Einstein \(n\)-manifold, \(n\geq 4,\) with positive scalar curvature \(s\) and of \(\text{Vol}(g)=1.\) Then, there exists a constant \(C(n)\), depending only on \(n\) such that if the \(L^{n/2}-\) norm satisfies \(\| W\|_{L^{n/2}}<C(n)s,\) then \(W=0\) so that \((M,g)\) is a finite isometric quotient of the standard \(n\)-sphere of unit volume. The idea of the proof is based on the Weitzenböck formula for the operator \( d_{L}:C^{\infty }(\Omega^{1}\otimes \Omega ^{2})\rightarrow C^{\infty }(\Omega ^{2}\otimes \Omega ^{2})\) and on the use of the Sobolev inequality relating to Yamabe metrics. A pointwise isolation theorem is similarly obtained for a compact connected oriented Einstein \(n\)-manifold \({(M,g)}\), \(n\geq 4,\) with positive scalar curvature \(s>0\). If \(\| W\|_{L^{n/2}}<(2/n)C_{n}s\) holds everywhere and strict inequality holds at a point, then \(W=0\), that is, \((M,g)\) is -- up to a constant scale -- a finite isometric quotient of the standard \(n\)-sphere.