Critical points of the total scalar curvature plus total mean curvature functional

The Total Scalar Curvature plus Total Mean Curvature functional is defined on the space of Riemannian metrics of a smooth compact manifold with boundary. We characterize its critical points restricted to spaces of Riemannian metrics satisfying various volume and area constraints, when the dimension of the manifold is n ≥ 3. In addition, we compute the second variation of said functional at critical points and exhibit directions in which it is positive, negative or zero. These results generalize to manifolds with boundary, well known results that hold in the case of manifolds without boundary.