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

TL;DR

This paper investigates the stability of submanifolds with constant higher order mean curvature within foliations, extending results on conformal fields and linking Jacobi fields to foliation-preserving vector fields.

Contribution

It generalizes recent results on conformal fields to arbitrary manifolds and explores the relationship between Jacobi fields and foliation-preserving vector fields.

Findings

Normal component of a Killing field is an rth Jacobi field for certain submanifolds.

Established links between rth Jacobi fields and foliation-preserving vector fields.

Extended conformal field results to more general manifold settings.

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.