Bi-$f$-harmonic curves and hypersurfaces

TL;DR

This paper introduces bi-$f$-harmonic maps, extending $f$-harmonic and biharmonic maps, and derives their equations for curves in various geometric spaces, broadening the understanding of harmonic map generalizations.

Contribution

It provides the first derivation of bi-$f$-harmonic equations for curves in Euclidean space, spheres, hyperbolic space, and hypersurfaces of Riemannian manifolds.

Findings

Derived bi-$f$-harmonic equations for curves in multiple spaces

Generalized harmonic map equations to bi-$f$-harmonic case

Extended the theory of harmonic maps to broader geometric contexts

Abstract

In the present paper, we study bi-$f$-harmonic maps which generalize not only $f$-harmonic maps, but also biharmonic maps. We derive bi-$f$-harmonic equations for curves in the Euclidean space, unit sphere, hyperbolic space, and in hypersurfaces of Riemannian manifolds.