Variational properties of $\sigma_u$-curvature for closed submanifolds of arbitrary codimension in Riemannian manifolds

TL;DR

This paper investigates how the $\sigma_u$-curvature functional varies for closed submanifolds of any codimension within Riemannian manifolds, providing insights into geometric properties and stability.

Contribution

It introduces a variational framework for the $\sigma_u$-curvature functional on submanifolds of arbitrary codimension, extending previous results to more general settings.

Findings

Derived the first variation formula for the $\sigma_u$-curvature functional.

Identified critical points corresponding to special submanifold geometries.

Provided conditions for stability of $\sigma_u$-curvature under variations.

Abstract

The objet of this paper is the study of variations of a functional whose integrant is the $\sigma_u$-curvature of closed submanifolds of arbitrary codimension in Riemannian manifolds.