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On critical Riemannian metrics for a curvature functional on 3-manifolds. (English) Zbl 1017.58009

One approach to geometrization problems in 3-manifolds \(M\) is to consider all Riemannian metrics \({\mathcal M}(M)\) on \(M\), to define a functional \(F: {\mathcal M}(M)\to\mathbb R\), and to pick out nice metrics on \(M\) as critical points of the functional. There are difficulties in the procedure: (1) Existence Theorems. Do there exist critical points? (2) Adequacy of the Functional. Does the functional control convergence or divergence of metrics? Of curvature of the metrics? Other properties of the metric?
The author considers as functional \(F\) a normalized \(L^2\)-norm of the traceless part of the Ricci curvature. He notes that Einstein metrics are critical points for his functional \(F\). In regard to existence of critical points for the functional, the author constructs a critical metric for \(F\) on the 3-sphere \(\mathbb S^3\) which is not Einstein. He proves that, if a metric is critical for \(F\) and satisfies an additional positivity condition, then the metric is Einstein.
Reviewer: J.W.Cannon (Provo)

MSC:

58E11 Critical metrics
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
57M50 General geometric structures on low-dimensional manifolds
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References:

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