On $f$-polyharmonic maps between Riemannian manifolds

This paper is devoted to a general study of $f$-polyharmonic maps of order $k$ (or $f$-$k$-harmonic maps), defined as critical points of the weighted $k$-energy functional $$E_{f,k}(\phi)=\frac{1}{2}\int_\Omega f |\overline{\Delta}^{k/2}\phi|^2 dv_g.$$ This framework provides a unifying perspective that extends previous theories including $f$-harmonic maps ($k=1$), biharmonic and $f$-biharmonic maps ($k=2$), and polyharmonic maps ($k\ge 3$ with constant $f$), with the classical harmonic maps recovered as the special case $k=1$ by setting $f\equiv \mathrm{const}$. We derive the Euler--Lagrange equation for general $f$-polyharmonic maps. As concrete applications, we classify $f$-$k$-harmonic curves with positive constant geodesic curvature in a space form $N^2(C)$ for $k=3,4$. Several explicit constructions of proper $f$-polyharmonic functions and maps are also provided, and a Liouville-type theorem is proved: every $f$-polyharmonic function on a closed Riemannian manifold is constant.