Some Remarks on Bi-f-Harmonic Maps and f-Biharmonic Maps

Abstract

In this paper, we prove that the class of bi-f -harmonic maps and that of fbiharmonic maps from a conformal manifold of dimension = 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 ≤ 1 everywhere (Theorems 2.2 and 3.3). A relationship between f -biharmonic maps and bi-f -harmonic mapsWe assume all objects studied in this paper, including manifolds, tension fields, and maps, are smooth unless they are stated otherwise. Harmonic maps are critical points of the energy functionalh) between Riemannian manifolds and for all compact domain Ω ⊆ M. Harmonic map equation is simply the Euler Lagrange equation of the energy functional which is given (see [ES]) by (1) τ (φ) ≡ Tr g ∇ dφ = 0, Date: 08/06/2018. 1991 Mathematics Subject Classification. 58E20, 53C43.

f-Biharmonic Submanifolds in Space Forms and f-Biharmonic Riemannian Submersions from 3-Manifolds

“…Proof. First of all, one sees that y = 0 is a solution of (8). For AC > 0, it is easy to check that (8) has constant solutions y = ± C A .…”

“…For AC > 0, it is easy to check that (8) has constant solutions y = ± C A . From now on, we only need to consider that y(s) is a nonconstant solution of (8). Putting y = u −1 and substituting this into (8), we obtain…”

“…The paper [4] shows that neither circular cone and part of the standard sphere S 2 in R 3 is f -biharmonic for any f , and a constant mean curvature surface in R 3 is f -biharmonic if and only if it is a part of a circular cylinder or a plane. For some recent progress on f -biharmonic submanifolds and some results on f -biharmonic maps, we refer the readers to [2,3,[5][6][7][8][9][10][11][12] and the references therein.…”