Constant k-curvature hypersurfaces in Riemannian manifolds

In [7], Rugang Ye (1991) proved the existence of a family of constant mean curvature hypersurfaces in an (m+1)(m+1)-dimensional Riemannian manifold (Mm+1,g)(Mm+1,g), which concentrate at a point p0p0 (which is required to be a nondegenerate critical point of the scalar curvature), moreover he proved that this family constitutes a foliation of a neighborhood of p0p0. In this paper we extend this result to the other curvatures (the r  -th mean curvature for 1⩽r⩽m1⩽r⩽m). And we give the expansion of the m  -dimensional volume of the leaves of this foliation as well as the (m+1)(m+1)-dimensional volume of the sets enclosed by each leaf.