Some remarks on bi-f-harmonic maps and f-biharmonic maps

TL;DR

This paper explores the relationship between bi-f-harmonic and f-biharmonic maps, establishing their equivalence under certain conditions and providing nonexistence results for proper maps into nonpositively curved manifolds.

Contribution

It proves the equivalence of bi-f-harmonic and f-biharmonic maps from conformal manifolds of dimension not equal to 2 and presents new nonexistence theorems for proper maps into nonpositively curved manifolds.

Findings

Bi-f-harmonic maps and f-biharmonic maps are equivalent from conformal manifolds of dimension ≠ 2.

Proper bi-f-harmonic and f-biharmonic maps into non-positively curved manifolds do not exist.

Maps with bounded f and energy from complete manifolds into negatively curved spaces have rank less than 2.

Abstract

In this paper, we prove that the class of bi-f-harmonic maps and that of f-biharmonic maps from a conformal manifold of dimension not equal to 2 are the same (Theorem 1.1). We also give several results on nonexistence of proper bi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifolds into nonpositively curved Riemannian manifolds. These include: any bi-f-harmonic map from a compact manifold into a non-positively curved manifold is f-harmonic (Theorem 1.6), and any f-biharmonic (respectively, bi-f-harmonic) map with bounded f and bounded f-bienrgy (respectively, bi-f-energy) from a complete Riemannian manifold into a manifold of strictly negative curvature has rank < 2 everywhere (Theorems 2.2 and 2.3).