Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems

Abstract

The Besse's conjecture was posted on the well-known book Einstein manifolds by Arthur L. Besse, which describes the critical point of Hilbert-Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse's conjecture and positive mass theorem for Brown-York mass. With the aid of positive mass theorem, we investigate the geometric structure of the so-called CPE manifolds, which provides us further understanding of Besse's conjecture. As a related topic, we also have a discussion of corresponding results for V -static metrics.

On the Mass of Static Metrics with Positive Cosmological Constant: II

“…We also mention that the rigidity statement for Theorem 1.5 will be deduced by some more general statements (see Corollaries 6.1, 6.5 and 8.7) which correspond to some balancing formulas, in the case where the boundary of N is allowed to have several connected components. The inequalities proven in Theorem 1.5 share some analogies with the ones developed in [26,55], see in particular [55,Theorem B].…”

Geometry of compact quasi-Einstein manifolds with boundary

“…Among the contributions that motivated this work, we primarily mention the classical isoperimetric inequality and a result due to Shen [35] and Boucher, Gibbons and Horowitz [11] that asserts that the boundary ∂M of a compact three-dimensional oriented triple static space (i.e., 1-quasi-Einstein manifold) with connected boundary and scalar curvature 6 must be a 2-sphere whose area satisfies the inequality |∂M | ≤ 4π, with equality if and only if M 3 is equivalent to the standard hemisphere. In the same spirit, boundary estimates for V -static metrics and static spaces were established in, e.g., [1,3,6,7,20,21,24,29,32]. In the recent work [22, Theorem 1], Diógenes and Gadelha proved an analogous boundary estimate for compact m-quasi-Einstein manifolds M n with connected boundary ∂M by assuming the following conditions:…”

Integral and boundary estimates for critical metrics of the volume functional

“…In spite of that, the isoperimetric constant obtained by them depends on the potential function f. It would be interesting to see if such a constant can be improved to depend only on the dimension and mean curvature of the boundary. Other boundary estimates for critical metrics were obtained in, e.g., [1,6,7,12,14].…”

“…Besides being interesting on their own, such estimates play a fundamental role in proving classification results and discarding some possible new examples of special metrics on a given manifold. In recent years, it was established some useful boundary and volume estimates for critical metrics of the volume functional, as for example, isoperimetric and Shen-Boucher-Gibbons-Horowitz type inequalities (see, e.g., [1,3,6,7,12,14]). At the same time, it is natural to explore integral estimates in order to obtain new obstruction results.…”