Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary

Abstract

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂ M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that M has divergence-free Bach tensor.

Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

“…To conclude this section, we shall present the following lemma for V -static spaces, which was previously obtained in [4] for Miao-Tam critical metrics. Lemma 1.…”

“…Corollaries 3.1 and 3.2 in [23]). For more details see, for instance, [3,4,5,12,22,23] and [31]. We also remark that Eq.…”

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

“…This tensor is closely tied to the Cotton tensor, and it played a fundamental role in the previous work [5] on classifying Bach-flat critical metrics of the volume functional. The tensor T ijk is also skew-symmetric in their first two indices and trace-free in any two indices.…”

“…In order to set the stage for the proof to follow let us recall an useful result obtained previously in [5,Lemma 1]. Lemma 1 ([5]).…”

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

“…where Ric g and Hess g f stand, respectively, for the Ricci tensor and the Hessian of f associated to g on M n (see [23] for more details). Hence, following the terminology used in [2,8,9,24] we recall the definition of Miao-Tam critical metric.…”