Constant $k$-curvature hypersurfaces in Riemannian manifolds

Abstract

Rugang Ye proved the existence of a family of constant mean curvature hypersurfaces in an $m+1$-dimensional Riemannian manifold $(M^{m+1},g)$, which concentrate at a point $p_0$ (which is required to be a nondegenerate critical point of the scalar curvature), moreover he proved that this family constitute a foliation of a neighborhood of $p_0$. In this paper we extend this result to the other curvatures (the $r$-th mean curvature for $1\le r\le m$).