The aim of this paper is to find some equations of structure for almost Ricci solitons which generalize the equivalent for Ricci solitons. As a consequence of these equations we derive an integral formula for the compact case which enables us to show that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal. Moreover, we also use the Hodge-de Rham decomposition theorem to make a link with the associated vector field of an almost Ricci soliton.

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.

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Abstract. We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manifolds. In addition, we obtain a Böchner type formula which enables us to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in S 3 .