F-biharmonic maps into general Riemannian manifolds

Abstract Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional EF,2(ψ)=∫MF|τ(ψ)|22dVg, $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C3 function such that F′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].


Introduction
In the past several decades harmonic map plays a central role in geometry and analysis. Harmonic maps between two Riemannian manifolds are critical points of the energy functional Here, dVg denotes the volume element of g. Harmonic maps equation is simply the Euler-Lagrange equation of the energy function E which is given (cf. [2]) by τ(ψ) = , where τ(ψ) is called the tension eld of ψ. By extending notion of harmonic map, in 1983, Eells and Lemaire [3] proposed to consider the bienergy functional E (ψ) = M |τ(ψ)| dVg *Corresponding Author: Rong Mi: College of Mathematics and Statistics, Northwest Normal University, Lanzhou, P. R. China; E-mail: mr8231227@163.com for smooth maps between two Riemannian manifolds. In 1986, Jiang [4,5] studied the rst and the second variational formulas of the bienergy functional and initiated the study of biharmonic maps. The Euler-Lagrange equation of E (ψ) is given by where ∆ ψ := m i= (∇e i∇ e i −∇ ∇ ei e i ), and∇ is the induced connection on the pullback ψ * TN, ∇ is the Levi-Civita connection on (M, g), and R N is the Riemannian curvature tensor on N. Clearly, any harmonic map is always a biharmonic map.

Conjecture 1.1. Every biharmonic submanifold of Euclidean space must be harmonic (minimal).
Here, if ψ : (M, g) → (N, h) is a biharmonic isometric immersion, then M is called a biharmonic submanifold in N.

Conjecture 1.2. Every biharmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic (minimal).
Indeed, biharmonic maps are not harmonic maps (the counter examples of [20]). It is interesting to nd some conditions to solve the biharmonic map equation which solutions reduce to harmonic map. About this results, there exists a large number of celebrated results (for instance, see [21 -26]). For harmonic maps, it is well know that: If a domain manifold (M, g) is complete and has non-negative Ricci curvature and the sectional curvature of a target manifold (N, h) is non-positive, then every nite harmonic map is a constant map [27]. See [28] and [29] for recent works on harmonic map. If M is noncompact, the maximum principle is no longer application. In this case we can use the integration by parts argument, by choosing proper test functions. Based on this idea, Baird, Fardoun and Ouakkas ( [30]) showed that: If a non-compact manifold (M, g) is complete and has non-negative Ricci curvature and (N, h) has non-positive sectional curvature, then every bienergy nite biharmonic map of (M, g) into (N, h) is harmonic. It is natural to ask whether we can abandon the curvature restriction on the domain manifold and weaken the integrability condition on the bienery. In this direction, Nakauchi et al. [8] showed that every biharmonic map of a complete Riemannian manifold into a Riemannian manifold of non-positive curvature whose bienergy and energy are nite must be harmonic. Later, Maeta [31] obtained that biharmonic maps from a complete Riemannian manifold into a non-positive curved manifold with nite p-bienergy M |τ(ψ)| p dVg < ∞ (p ≥ ) and energy are harmonic. In [14], Han and Zhang investigated harmonicity of p-biharmonic maps, as the cases of F-biharmonic maps. Moreover, in [15] Han introduced the notion of p-biharmonic submanifold and proved that p-biharmonic submanifold (M, g) in a Riemannian manifold (N, h) with non-positive sectional curvature which satis es certain condition must be minimal. Furthermore, Luo in [32,33] respectively generalized these results.
Recently, Branding and Luo [26] proved a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds, by assuming that the sectional curvature of the Riemannian manifold have an upper bound and give a more natural integrability condition, which generalized Branding's. It is natural to ask whether we can generalize the results on the F-biharmonic maps. In [7], Han and Feng proved that F-biharmonic map from a compact orientable Riemannian manifold into a Riemannian manifold with non-positive sectional curvature are harmonic. There have been extensive studies in this area (for instance, [34,35]). In the following, we aim to establish another nonexistence result for F-biharmonic maps that does not require any assumption on the curvature of the target manifold, instead we will demand that the tension eld is small in a certain L p -norm and satis es m-energy niteness. To state our theorem, we rst give a de nition.  [38]). Here, ωm denotes the volume of the unit ball in R m . For a discussion of this kind of problem, we suggest the reader refer to [36].

De nition 1.1. [36, 37] An m-dimensional (m ≥ ) complete non-compact Riemannian manifold M of in nite volume admits an Euclidean type Sobolev inequality
Motivated by these aspects, we actually can prove the following result:  [8]), biharmonic maps (cf. [1]) and [39] for a more general result.
Furthermore, we can get the following Liouville-type result.

Lemmas
In order to prove our theorems, we need the following lemmas. For a xed point x ∈ M and for every R > , let us consider the following cut-o function η(x) on M: Proof. Multiplying the F-biharmonic map equation by a test function of the term: where p > and δ > , then we have (2.1)

Integrating over M and using integration by parts we get
We divide the arguments into two cases: -Case 1: When < p < , we obtain from (2.2) that where the inequality follows from

Using (2.1) and (2.3) we deduce that
which proves the rst claim.
-Case 2: When p ≥ , from (2.2) we get By a straightforward computation we can rewrite (2.4) as This completes the proof of Lemma 2.1.

Lemma 2.2. Assume that (M, g) satis es the assumptions of Theorem 1.1, then the following inequality holds
By using Hölder's inequality, we have (2.5) (2.6) Thus, we obtain the Lemma 2.2.
In the following we will make use of the following result due to Ga ney [41].

Proof of the main result
In this section we will give a proof of Theorem 1.1 .
Proof. We divide the arguments into two cases: - where C is a constant depending on p, K and the geometry of M. Now, let M = {x ∈ M|F ( |τ(ψ)| )τ(ψ)(x) = }, and M = M \ M . If M is an empty set, this completes the proof. Hence, we assume that M is nonempty and we will get a contradiction below. Since ψ is smooth, M is an open set, we use (3.1) to have Letting δ → we have Letting R → ∞ we get -Case 2: When p ≥ , by a similar discussion we can prove that