On $f$-bi-harmonic maps between Riemannian manifolds

TL;DR

This paper introduces and studies $f$-bi-harmonic maps, a new class combining bi-harmonic and $f$-harmonic maps, deriving their equations and analyzing their properties in various geometric contexts.

Contribution

It defines $f$-bi-harmonic maps as critical points of a new energy functional, generalizing harmonic and bi-harmonic maps, and derives their governing equations.

Findings

Derived the $f$-bi-harmonic map equation.

Analyzed $f$-bi-harmonicity of conformal maps.

Studied $f$-bi-harmonicity of product and projection maps.

Abstract

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy functional $\frac{1}{2}\int_M f|\tau (\phi)|^2dv_{g}$. This class of maps generalizes both concepts of harmonic maps and bi-harmonic maps. We first derive the $f$-biharmonic map equation and then use it to study $f$-bi-harmonicity of some special maps, including conformal maps between manifolds of same dimensions, some product maps between direct product manifold and singly warped product manifold, some projection maps from and some inclusion maps into a warped product manifold.