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

By A. Barros, R. Diógenes, E. Ribeiro. 2014

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The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $$M$$M with boundary $$\partial M$$∂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 $$\mathbb {R}^{4}, \mathbb {H}^{4}$$R4,H4 or $$\mathbb {S}^{4}$$S4. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that $$M$$M has divergence-free Bach tensor.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/98ed1aba1e752180b44fd81c279065b526237c70