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Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature. (English) Zbl 0881.53036
Let \(M\) be a compact smooth manifold of dimension \(n\geq 3\). The Yamabe functional \(I\) on the conformal class \(C\) of \(M\) is defined by \(I(g)= {\int_MS_g dv_g \over V_g^{(n-2)/n}}\) for \(g\in C\), where \(S_g\), \(dv_g\) and \(V_g\) denote the scalar curvature, the volume element and the volume of \((M,g)\), respectively. The Yamabe invariant of \((M,C)\) is the infimum of this functional and is denoted by \(\mu(M,C)\). A metric \(g\) which minimizes the Yamabe functional \(I\) on the conformal class \(C\) is called a Yamabe metric.
Let \({\mathcal Y}_1 (n,\mu_0)\) be the class of compact connected smooth \(n\)-manifolds \(M(n\geq 3)\) with Yamabe metrics \(g\) of unit volume which satisfy \(\mu(M,[g]) \geq\mu_0>0\), where \([g]\) denotes the conformal class of \(g\). In the paper under review, the author proves several convergence theorems for Riemannian manifolds in \({\mathcal Y}_1 (n,\mu_0)\) with integral bounds on curvature. One of them includes a pinching theorem for flat conformal structures of positive Yamabe invariant on compact 3-manifolds.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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