We provide a general Böchner type formula which enables us to prove some rigidity results for V -static spaces. In particular, we show that an n-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify V -static spaces with non-negative sectional curvature.

Abstract. The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected n-dimensional manifold M n with boundary ∂M has parallel Ricci tensor, then M n is isometric to a geodesic ball in a simply connected space form R n , H n or S n .

Let C be the space of smooth metrics g on a given compact manifold M n (n ≥ 3) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space C (we shall refer to this critical point as CPE metrics) under assumption that (M, g) has zero radial Weyl curvature.Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard 3-sphere. We will also prove that a n-dimensional, 4 ≤ n ≤ 10, CPE metric satisfying a L n/2 -pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.

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We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics. Date: December 1, 2017. 2010 Mathematics Subject Classification. Primary 53C25, 53C20, 53C21; Secondary 53C65. Key words and phrases. Volume functional; critical metrics; isoperimetric inequality; Weitzenböck formula. H. Baltazar was partially supported by CNPq/Brazil and FAPEPI/Brazil. E. Ribeiro Jr was partially supported by grants from CNPq/Brazil (Grant: 303091/2015-0), PRONEX-FUNCAP/CNPq/Brazil and CAPES/ Brazil -Finance Code 001.