Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric

Critical metrics of the volume functional with pinched curvature

“…There are several works about classification of Miao-Tam critical metrics involving the Weyl curvature tensor, see for instance, [3,4,5,7,9,17]. Hence, motivated by [12] and based in techniques developed in [1,2], we shall provide a classification of a such critical metrics considering a pointwise pinching condition.…”

Critical metrics on $4$-manifolds with harmonic anti-self dual Weyl tensor

“…This result includes, in particular, the half-conformally flat case (i.e., W + = 0). Recently, Kim and Shin [12] showed that a simply connected, compact Miao-Tam critical metric with harmonic Weyl tensor (i.e., div(W ) = 0) 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 (see also [2,4] for an alternative proof).…”

Remarks on compact quasi-Einstein manifolds with boundary

“…In another direction, Catino, Mastrolia and Monticelli [19] obtained an interesting classification result for gradient Ricci solitons admitting a fourth-order vanishing condition on the Weyl tensor. To be precise, they showed that any (n ≥ 4)dimensional gradient shrinking Ricci soliton with fourth-order divergence-free Weyl tensor (i.e., div 4…”

“…Notice that the fourth-order divergence-free Weyl tensor assumption is clearly weaker than locally conformally flat and harmonic Weyl tensor conditions considered in [20,26]. Moreover, this assumption was recently used to classify critical metrics of the volume functional, static spaces and CPE metrics, for more details see, e.g., [3,4,19,35,38]. It is also important to emphasize that Example 1 has fourth-order divergence-free Weyl tensor, but it has no zero radial Weyl curvature.…”

“…Let M n , g, u, λ , n ≥ 4, be a nontrivial compact simply connected (m = 1)-quasi-Einstein manifold with smooth boundary ∂M and zero radial Weyl curvature. Suppose that M n has fourth-order divergence-free Weyl tensor (i.e., div 4 (W ) = 0). Then M n , g, u) is either…”