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On critical point equation of compact manifolds with zero radial Weyl curvature

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Abstract

Let \(\mathcal {C}\) be the space of smooth metrics g on a given compact manifold \(M^{n}\) (\(n\ge 3\)) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space \(\mathcal {C}\) (we shall refer to this critical point as CPE metrics) under assumption that (M, g) has zero radial Weyl curvature. Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard 3-sphere. We will also prove that a n-dimensional, \(4\le n\le 10,\) CPE metric satisfying a \(L^{n/2}\)-pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.

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Acknowledgements

The author want to thank M. Matos Neto for helpful discussions about this subject.

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  1. Departamento de Matemática, Universidade Federal do Piauí, Teresina, Piauí, 64049-550, Brazil

    H. Baltazar

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  1. H. Baltazar
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Baltazar, H. On critical point equation of compact manifolds with zero radial Weyl curvature. Geom Dedicata 202, 337–355 (2019). https://doi.org/10.1007/s10711-018-0417-3

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