  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.
