F-biharmonic maps into general Riemannian manifolds

Rong Mi

doi:10.1515/math-2019-0112

Artikel Open Access

F-biharmonic maps into general Riemannian manifolds

Rong Mi

Veröffentlicht/Copyright: 8. November 2019

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 17 Heft 1

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,

where F : [0, ∞) → [0, ∞) be C³ function such that F^′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τ_F,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 L^p-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].

Keywords: Sobolev inequality; Lp-norm; harmonic

MSC 2010: 58E20; 53C43

1 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

E(ψ)=12∫M|dψ|2dVg,

for smooth maps ψ : (M, g) → (N, h) between Riemannian manifolds and for all compact domain Ω ⊆ M. Here, dV_g 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) = 0, where τ(ψ) is called the tension field of ψ. By extending notion of harmonic map, in 1983, Eells and Lemaire [3] proposed to consider the bienergy functional

E2(ψ)=12∫M|τ(ψ)|2dVg

for smooth maps between two Riemannian manifolds. In 1986, Jiang [4, 5] studied the first and the second variational formulas of the bienergy functional and initiated the study of biharmonic maps. The Euler-Lagrange equation of E₂(ψ) is given by

τ2(ψ):=−Δψτ(ψ)−∑i=1mRNτ(ψ),dψ(ei)dψ(ei)=0,

where Δψ:=∑i=1m(∇¯ei∇¯ei−∇¯∇eiei), 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.

Later, Ara [6], Han and Feng [7] introduced the F-bienergy functional

EF,2(ψ)=∫MF|τ(ψ)|22dVg,

where F : [0, ∞) → [0, ∞) be C³ function such that F^′ > 0 on (0, ∞). The map ψ is called an F-biharmonic map if it is a critical point of that F-bienergy E_F,2(ψ), which is a generalization of biharmonic maps (cf. [8, 9, 10, 11, 12]), p-biharmonic maps (cf. [13, 14, 15, 16]) or exponentially biharmonic maps (cf. [7]). The Euler-Lagrange equation of E_F,2(ψ) is

τF,2(ψ):=−ΔψF′(|τ(ψ)|22)τ(ψ)−∑i=imRNF′(|τ(ψ)|22)τ(ψ),dψ(ei)dψ(ei)=0.

A map ψ : (M, g) → (N, h) is called a F-biharmonic map if τ_F,2(ψ) = 0. Note that harmonic maps are always F-biharmonic by definition.

When F(t) = e^t, we have exponential bienergy functional Ee,2(ψ)=∫Me|τ(ψ)|22dVg.

When F(t)=(2t)p2, we have p-bienergy functional E_p,2(ψ) = ∫_M|τ(ψ)|^p dV_g, where p ≥ 2. The Euler-Lagrange equation of E_p,2 is

τp,2(ψ):=−Δψp|τ(ψ)|p−2τ(ψ)−∑i=1mRNdψ(ei),p|τ(ψ)|p−2τ(ψ)dψ(ei)=0.

A map ψ : (M, g) → (N, h) is called a p-biharmonic map if τ_p,2(ψ) = 0.

B. Y. Chen [17] raised so called B. Y. Chen’s conjecture and later, [18] and [19] raised the generalized B. Y. Chen’s conjecture.

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 find 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, 22, 23, 24, 25, 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 finite 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 finite 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 finite must be harmonic. Later, Maeta [31] obtained that biharmonic maps from a complete Riemannian manifold into a non-positive curved manifold with finite p-bienergy ∫_M|τ(ψ)|^p dV_g < ∞ (p ≥ 2) 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 satisfies 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 field is small in a certain L^p-norm and satisfies m-energy finiteness. To state our theorem, we first give a definition.

Definition 1.1

[36, 37] An m-dimensional (m ≥ 3) complete non-compact Riemannian manifold M of infinite volume admits an Euclidean type Sobolev inequality

∫M|u|2mm−2dVgm−2m≤CSobM∫M|∇u|2dVg, (1.1)

for all u ∈ W^1,2(M) with compact support, where CSobM is a positive constant that depends on the geometry of M.

Remark 1.1

Such an inequality holds in ℝ^m and is well-known as Gagliardo-Nirenberg inequality in this case.

Remark 1.2

Suppose that (M, g) is a complete non-compact Riemannian manifold of dimension m = dimM with nonnegative Ricci curvature. Suppose that for some point x ∈ M,

limR→∞Volg(BR(x))ωmRm>0

holds, then (1.1) holds true (cf. [38]). Here, ω_m 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].

Motivated by these aspects, we actually can prove the following result:

Theorem 1.1

Suppose that (M, g) is a complete, connected non-compact Riemannian manifold of m = dimM ≥ 3 with infinite volume that admits an Euclidean type Sobolev inequality of the form (1.1). Moreover, suppose that (N, h) is another Riemannian manifold whose sectional curvature satisfy K^N ≤ K, where K is a positive constant. Let ψ : (M, g) → (N, h) be a smooth F-biharmonic map. If

∫MF′(|τ(ψ)|22)τ(ψ)pdVg<∞

and

∫M|dψ|mdVg<ϵ,

for p > 1 and ϵ > 0 (depending on p, K, and the geometry of M) sufficiently small, then ψ must be harmonic.

Remark 1.3

For a better understand of Theorem 1.1, the readers could consult the papers for examples of p-harmonic maps (cf. [8]), biharmonic maps (cf. [1]) and [39] for a more general result.

Furthermore, we can get the following Liouville-type result.

Corollary 1.1

Suppose that (M, g) is a complete, connected non-compact Riemannian manifold of m = dimM ≥ 3 with nonnegative Ricci curvature that admits an Euclidean type Sobolev inequality of the form (1.1). Moreover, suppose that (N, h) is another Riemannian manifold whose sectional curvature satisfy K^N ≤ K, where K is a positive constant. Let ψ : (M, g) → (N, h) be a smooth F-biharmonic map. If

∫MF′(|τ(ψ)|22)τ(ψ)pdVg<∞

and

∫M|dψ|mdVg<ϵ,

for p > 1 and ϵ > 0 (depending on p, K, and the geometry of M) sufficiently small, then ψ is a constant map.

Remark 1.4

Note that due to a classical result of Yau [40], a complete noncompact Riemannian manifold with nonnegative Ricci curvature has infinite volume.

2 Lemmas

In order to prove our theorems, we need the following lemmas.

For a fixed point x₀ ∈ M and for every R > 0, let us consider the following cut-off function η(x) on M:

0≤η(x)≤1,x∈M;η(x)=0,x∈M∖B2R(x0);η(x)=1,x∈BR(x0);|∇η(x)|≤CR,x∈M,

where B_R(x₀) = {x ∈ M : d(x, x₀) < R}, C is a positive constant and d is the distance on (M, g).

Lemma 2.1

Let ψ : (M, g) → (N, h) be a smooth F-biharmonic map and assume the sectional curvature of N satisfy K^N ≤ K, where K is a positive constant. Let δ be a positive constant, then the following inequality holds:

If 1 < p < 2, we get

1−p−12∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg+K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg.
If p ≥ 2, we get

12∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg.

Proof

Multiplying the F-biharmonic map equation by a test function of the term:

η2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22F′(|τ(ψ)|22)τ(ψ),

where p > 1 and δ > 0, then we have

η2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22ΔψF′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)=−η2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∑i=1mRNF′(|τ(ψ)|22)τ(ψ),dψ(ei),F′(|τ(ψ)|22)τ(ψ),dψ(ei). (2.1)

Integrating over M and using integration by parts we get

∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22ΔψF′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg=−2∫Mη∇η[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)2dVg−∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∇¯F′(|τ(ψ)|22)τ(ψ)2dVg≤−2∫Mη∇η[F′(|τ(ψ)|22)τ(ψ)]2+δ)p−22∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg−(p−2)η2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)2dVg−∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg. (2.2)

We divide the arguments into two cases:

When 1 < p < 2, we obtain from (2.2) that

∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22ΔψF′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg≤(p−12−1)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg+2p−1∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|∇η|2dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)2dVg≤p−12−1∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg, (2.3)

where the inequality follows from

∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)2≤∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2.

Using (2.1) and (2.3) we deduce that

1−p−12∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∑i=1mRNF′(|τ(ψ)|22)τ(ψ),dψ(ei),F′(|τ(ψ)|22)τ(ψ),dψ(ei)dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22F′(|τ(ψ)|22)τ(ψ)2|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg−(p−2)∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg,

which proves the first claim.
When p ≥ 2, from (2.2) we get

∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22ΔψF′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg≤−2∫Mη∇η[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)dVg−∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤−12∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg+2∫M[F′(|τ(ψ)|22)τ(ψ)]2+δp2|∇η|2dVg. (2.4)

By a straightforward computation we can rewrite (2.4) as

12∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22∑i=1mRNF′(|τ(ψ)|22)τ(ψ),dψ(ei),F′(|τ(ψ)|22)τ(ψ),dψ(ei)dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg≤K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−22F′(|τ(ψ)|22)τ(ψ)2|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg≤K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg+CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg.

This completes the proof of Lemma 2.1. □

Next, we deal with the term K∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg.

Lemma 2.2

Assume that (M, g) satisfies the assumptions of Theorem 1.1, then the following inequality holds

∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg≤2C2⋅CSobM∫M|dψ|mdVg2m⋅1R2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg+p2⋅CSobM2∫M|dψ|mdVg2m×∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg.

Proof

Set f=F′(|τ(ψ)|22)τ(ψ)+δp4, then we get

∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp2|dψ|2dVg=∫Mη2|dψ|2f2dVg.

By using Hölder’s inequality, we have

∫Mη2|dψ|2f2dVg≤∫M(ηf)2mm−2dVgm−2m∫M|dψ|mdVg2m. (2.5)

Applying (1.1) to u = η f we get

∫M(ηf)m−2mdVg2mm−2≤CSobM∫M|d(ηf)|2dVg. (2.6)

Substituting (2.6) into (2.5) yields

∫Mη2|dψ|2f2dVg≤2CSobM∫M|dψ|mdVg2m∫M|dη|2f2dVg+∫M|η|2|df|2dVg.

Using f=[F′(|τ(ψ)|22)τ(ψ)]2+δp4 and

|df|2=p24[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ),F′(|τ(ψ)|22)τ(ψ)2≤p24[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2.

Thus, we obtain the Lemma 2.2. □

In the following we will make use of the following result due to Gaffney [41].

Lemma 2.3

∫MδωdVg=∫MdivXdVg=0.

3 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:

When 1 < p < 2, from Lemma 2.1 and Lemma 2.2, we can choose ϵ sufficiently small such that Kp2⋅CSobM2ϵ2m≤p−14, a straightforward computation shows that

p−14∫Mη2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg, (3.1)

where C is a constant depending on p, K and the geometry of M. Now, let M₁ = {x ∈ M|F′(|τ(ψ)|22)τ(ψ)(x) = 0}, and M₂ = M ∖ M₁. If M₂ is an empty set, this completes the proof. Hence, we assume that M₂ is nonempty and we will get a contradiction below. Since ψ is smooth, M₂ is an open set, we use (3.1) to have

p−14∫M2η2[F′(|τ(ψ)|22)τ(ψ)]2+δp−42+∇¯F′(|τ(ψ)|22)τ(ψ)2F′(|τ(ψ)|22)τ(ψ)2dVg≤CR2∫B2R(x0)[F′(|τ(ψ)|22)τ(ψ)]2+δp2dVg.

Letting δ → 0 we have

p−14∫M2η2F′(|τ(ψ)|22)τ(ψ)p−2∇¯F′(|τ(ψ)|22)τ(ψ)2≤CR2∫B2R(x0)F′(|τ(ψ)|22)τ(ψ)pdVg≤CR2∫MF′(|τ(ψ)|22)τ(ψ)pdVg.

Letting R → ∞ we get

p−14∫M2F′(|τ(ψ)|22)τ(ψ)p−2∇¯F′(|τ(ψ)|22)τ(ψ)2dVg=0.
When p ≥ 2, by a similar discussion we can prove that

p−14∫M2F′(|τ(ψ)|22)τ(ψ)p−2∇¯F′(|τ(ψ)|22)τ(ψ)2dVg=0.

Therefore we conclude that ∇¯F′(|τ(ψ)|22)τ(ψ)=0 everywhere in M₂. It follows that M₂ is an open and closed nonempty set, hence, M₂ = M and |F′(|τ(ψ)|22)τ(ψ)|≡c for some constant c ≠ 0. Thus Vol(M) < ∞ by ∫_M c^p d V_g < ∞.

Next, making using of Gaffney’s result (Lemma 2.3). Define a 1-form ω on M as follows

ω(X)=dψ(X),F′(|τ(ψ)|22)τ(ψ),∀X∈TM.

Then we have

∫M|ω|dVg=∫M∑i=1m|ω(ei)|212dVg≤∫MF′(|τ(ψ)|22)τ(ψ)|dψ|dVg≤c(VolM)1−1m∫M|dψ|mdVg1m<∞.

Now, we compute −δω=∑i=1m∇eiω(ei) to have

−δω=∑i=1m∇eiω(ei)−ω(∇eiei)=∑i=1m∇¯eidψ(ei),F′(|τ(ψ)|22)τ(ψ)−dψ(∇eiei),F′(|τ(ψ)|22)τ(ψ)=∑i=1m∇¯eidψ(ei)−dψ(∇eiei),F′(|τ(ψ)|22)τ(ψ)=F′(|τ(ψ)|22)τ(ψ)2,

where in obtaining the second equality we have used ∇¯F′(|τ(ψ)|22)τ(ψ)=0. Therefore,

∫M|δω|dVg=c2⋅Vol(M)<∞.

Now by Gaffney’s theorem and the above equality, we have

0=∫M(−δω)dVg=∫MF′(|τ(ψ)|22)τ(ψ)2dVg=c2⋅Vol(M),

which implies that c = 0, a contradiction. Hence, we have M₁ = M and ψ is a harmonic map. This completes the proof of Theorem 1.1. □

Acknowledgements

The author would like to thank the referee for his/her comments and suggestions.

References

[1] Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, arXiv: 1806.11441v2.Suche in Google Scholar

[2] Eells J., Sampson J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math., 1964, 86, 109–160.10.1142/9789814360197_0001Suche in Google Scholar

[3] Eells J., Lemaire L., Selected topics in harmonic maps, Amer. Math. Soc. CBMS, 1983, 50.10.1090/cbms/050Suche in Google Scholar

[4] Jiang G. Y., 2-Harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A(7), 1986, 4, 389–402.Suche in Google Scholar

[5] Jiang G. Y., 2-Harmonic isometric immersion between Riemannian manifolds, Chinese Ann. Math. Ser. A(7), 1986, 2, 130–144.Suche in Google Scholar

[6] Ara M., Geometry of F-harmonic maps, Kodai Math. J., 1999, 22, 243–263.10.2996/kmj/1138044045Suche in Google Scholar

[7] Han Y. B., Feng S. X., Some results of F-biharmonic maps, Acta Math. Univ. Comenianae, 2014, 1, 47–66.Suche in Google Scholar

[8] Nakauchi N., Urakawa H., Gudmundsson S., Biharmonic maps in a Riemannian manifold of non-positive curvature, Geom. Dedicata, 2014, 164, 263–272.10.1007/s10711-013-9854-1Suche in Google Scholar

[9] Chiang Y. J., Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields. Basel: Springer Basel, 2013.10.1007/978-3-0348-0534-6Suche in Google Scholar

[10] Loubeau E., The index of biharmonic maps in spheres, Compositio. Math., 2005, 141, 729–745.10.1112/S0010437X04001204Suche in Google Scholar

[11] Wang C. Y., Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations, 2004, 21, 221–242.10.1007/s00526-003-0252-7Suche in Google Scholar

[12] Wang C. Y., Biharmonic maps from ℝ⁴ into Riemannian manifold, Math. Z., 2004, 247, 65–87.10.1007/s00209-003-0620-1Suche in Google Scholar

[13] Hornung P., Moser R., Intrinsically p-biharmonic maps, Preprint (opus: University of Bath online publication store).10.1007/s00526-013-0688-3Suche in Google Scholar

[14] Han Y. B., Some results of p-biharmonic submanifolds in a Riemannian manifold of non-positive curvature, J. Geom., 2015, 106, 471–482.10.1007/s00022-015-0259-1Suche in Google Scholar

[15] Han Y. B., Zhang W., Some results of p-biharmonic maps into non-positively curved manifold, J. Korean Math. Soc., 2015, 52, 1098–1108.10.4134/JKMS.2015.52.5.1097Suche in Google Scholar

[16] Cao X. Z., Luo Y., On p-biharmonic submanifolds in non-positively curved manifold, Kodai Math. J., 2016, 39(3), 567–578.10.2996/kmj/1478073773Suche in Google Scholar

[17] Chen B. Y., Some open problems and conjectures on submanifolds of finite type, Michigan State University, 1988.Suche in Google Scholar

[18] Balmus A., Montaldo S., Onicius C., Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr., 2010, 283, 1696–1705.10.1002/mana.200710176Suche in Google Scholar

[19] Caddeo R., Montaldo S., Onicius C., Biharmonic submanifolds of 𝕊³, Int. J. Math., 2001, 12, 867–876.10.1142/S0129167X01001027Suche in Google Scholar

[20] Loubeau E., Ou Y. L., Biharmonic maps and morphisms from conformal mappings, Tohoku Math. J., 2010, 62, 55–73.10.2748/tmj/1270041027Suche in Google Scholar

[21] Montaldo S., Oniciuc C., A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina, 2007, 47, 1–22.Suche in Google Scholar

[22] Luo Y., Liouville-type theorems on complete manifolds and non-existence of bi-harmonic maps, J. Geom. Anal., 2014, 10.1007/s12220-01409521-2.Suche in Google Scholar

[23] Chiang Y. J., Developments of harmonic maps, wave maps and Yang-Mills fields into biharmonic maps, biwave maps and Bi-Yang-Mills fields, Front. Math., 2013.10.1007/978-3-0348-0534-6Suche in Google Scholar

[24] Loubeau E., Oniciuc C., The index of biharmonic maps in spheres, Compositio Math., 2005, 141, 729–745.10.1112/S0010437X04001204Suche in Google Scholar

[25] Loubeau E., Montaldo S., Oniciuc C., The stress-energy tensor for biharmonic maps, Math. Z., 2008, 259, 503–524.10.1007/s00209-007-0236-ySuche in Google Scholar

[26] Branding V., A Liouville-type theorem for proper biharmonic maps between complete Riemannian manifolds with small energies, arXiv: 1712.03870v2.Suche in Google Scholar

[27] Schoen R., Yau S.T., Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature, Comment Math. Helv., 1976, 51, 333–341.10.1007/BF02568161Suche in Google Scholar

[28] Inoguchi J., Submanifolds with harmonic mean curvature vector field in contact 3-manifolds, Colloq. Math., 2004, 100, 163–179.10.4064/cm100-2-2Suche in Google Scholar

[29] Kajigaya T., Second variation formula and the stability of Legendrian minimal submanifolds in Sasakian manifolds, Diff. Geom. Appl., 2014, 37, 89–108.10.2748/tmj/1386354294Suche in Google Scholar

[30] Baird P., Fardoun A., Ouakkas S., Liouville-type theorems for biharmonic maps between complete Riemannian manifolds, Adv. Calc. Var., 2010, 3, 49–68.10.1515/acv.2010.003Suche in Google Scholar

[31] Maeta S., Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold, Ann. Glob. Anal. Geom., 2014, 46, 75–85.10.1007/s10455-014-9410-8Suche in Google Scholar

[32] Luo Y., The maximal principle for properly immersed submanifolds and its applications, Geom. Dedicata, 2016, 181, 103–112.10.1007/s10711-015-0114-4Suche in Google Scholar

[33] Luo Y., Remarks on the nonexistence of biharmonic maps, Arch. Math. (Basel), 2016, 107, 191–200.10.1007/s00013-016-0924-0Suche in Google Scholar

[34] Dong Y. X., Lin H. Z., Yang G. L., Liouville-type theorems for F-harmonic maps and their applications, Results. Math., 2016, 69, 105–127.10.1007/s00025-015-0480-0Suche in Google Scholar

[35] Kassi M., A Liouville-type theorems for F-harmonic maps with finite F-energy, Electronic Journal Differential Equations, 2006, 15, 1–9.Suche in Google Scholar

[36] Hebey E., Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, 1635, Springer-Velag, Berlin, 1996.10.1007/BFb0092907Suche in Google Scholar

[37] Zhen S., Some rigidity phenomena for Einstein metrics, Proc. Amer. Math. Soc., 1990, 108, 981–987.10.1090/S0002-9939-1990-1007511-2Suche in Google Scholar

[38] Shen Z., Some rigidity phenomena for Einstein metrics, Proc. Amer. Math. Soc., 1990, 108, 981–987.10.1090/S0002-9939-1990-1007511-2Suche in Google Scholar

[39] Zhou Z., Chen Q., Global pinching lemmas and their applications to geometry of submanifolds, harmonic maps and Yang-Mills fields, Adv. Math. (China), 2003, 32, 319–326.Suche in Google Scholar

[40] Yau S., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 1976, 25, 659–670.10.1512/iumj.1976.25.25051Suche in Google Scholar

[41] Gaffney M. P., A special Stokes’s theorem for complete Riemannian manifolds, Ann. of Math., 1954, 60, 140–145.10.2307/1969703Suche in Google Scholar

Received: 2019-04-15

Accepted: 2019-09-27

Published Online: 2019-11-08

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2019-0112

Schlagwörter für diesen Artikel

Sobolev inequality; Lp-norm; harmonic

Creative Commons

BY 4.0