Four dimensional static and related critical spaces with harmonic curvature

Volume functional of compact $4$-manifolds with a prescribed boundary metric

“…which is clearly weaker than locally conformally flat and harmonic Weyl tensor conditions considered in [23] and [19]. More precisely, we have established the following result.…”

“…Proceeding, since M 4 has second order divergence-free Weyl tensor, we may use (3.7) jointly with (2.7) to deduce that the Cotton tensor C is identically zero in M 4 . This implies that the Weyl tensor is harmonic, and then it suffices to apply Theorem 10.3 in [19] to conclude that M 4 is isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . This finishes the proof of Theorem 1.…”

“…The Bach-flat condition can be seen as a vanishing condition involving second and zero order terms in the Weyl tensor. Recently, Kim and Shin [19] showed that a simply connected, compact Miao-Tam critical metric with harmonic curvature and boundary isometric to a standard sphere S 3 must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . But, since a Miao-Tam critical metric has constant scalar curvature, we conclude that the assumption of harmonic curvature in the Kim-Shin result can be replaced by the harmonic Weyl tensor condition (i.e.…”

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

“…Meanwhile, Barros, and Da Silva presented in [6], an upper bound for the area of the boundary of a compact n-dimensional oriented Miao-Tam critical metric (see also [5,9]). For more references on the critical metrics of the volume functional, see [2,3,6,7,9,20,25], and references therein.…”

On the volume functional of compact manifolds with boundary with harmonic Weyl tensor

“…[18,Corollary 4.1]) by the Bachflat condition. Later on, Kim and Shin [14] also in 4-dimensional case, proved that a simply connected, compact Miao-Tam critical metric with boundary isometric to a standard sphere S 3 must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 , provided that the manifold has harmonic curvature. At this point it is important to remember that, if a Riemannian manifold with constant scalar curvature has harmonic curvature, then its Weyl tensor must be harmonic (the converse of this fact is also true for manifold with constant scalar curvature).…”