Conformal fields and the stability of leaves with constant higher order mean curvature

Abstract

In this paper, we study submanifolds with constant $r$th mean curvature $S_r$. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Al\'{i}as - Colares, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is a $r$th Jacobi field of a submanifold with $S_{r+1}$ constant. Finally, we study relations between $r$th Jacobi fields and vector fields preserving a foliation.