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Keywords:
conformally flat; 4-manifold; variational characterization
Summary:
In this paper, we give a new variational characterization of certain 4-manifolds. More precisely, let $R$ and $Ric$ denote the scalar curvature and Ricci curvature respectively of a Riemannian metric, we prove that if $(M^{4},g)$ is compact and locally conformally flat and $g$ is the critical point of the functional $$F(g)=\int _{M^{4}}(aR^{2}+b|Ric|^{2})\,\mathrm {d}v_{g}\,,$$ where $$(a,b)\in \mathbb {R}^{2}\setminus L_{1}\cup L_{2}$$ $$L_{1}\colon 3a+b=0\,;\quad L_{2}\colon 6a-b+1=0\,,$$ then $(M^{4},g)$ is either scalar flat or a space form.
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