arXiv:1806.11441v2 [math.DG] 13 Jul 2018 field emty nldn iia ufc hoy ope geome complex theory, surface minimal including geometry, as n18,ElsadLmie(4,seas 5)proposed [5]) also see ([4], Lemaire and Eells 1983, In maps. as oe iiieso h inryfntoa.I 96 Jiang of initia 1986, and equation In functional Euler-Lagrange bienergy functional. the of bienergy formulas variational the of minimizers p more, Stationary manifolds. called Riemannian are between maps smooth for where h ue-arneeuto of equation Euler-Lagrange The o a For ( Let Introduction 1 oeitnetermfrpoe iamncmp nogene into maps biharmonic proper for theorem nonexistence A uheothsbe adi h atsvrldcdst general to decades several last the in paid been has effort Much applications important many has maps harmonic of theory The ,g M, of o-opc imninmnflso dimension of manifolds Riemannian non-compact di nEcientp ooe nqaiyit eea imninman Riemannian general into of inequality finiteness Sobolev type Euclidean an admit eas e eea oeitnerslsfrpoe iamncsu biharmonic manifolds. proper Riemannian for general results into manifolds nonexistence non-compact several get also we eetrsl ftefis ae uhr hr easmd2 assumed he where author, named first the of result recent a W φ ∗ φ 1 nti oew rv oeitnersl o rprbhroi m biharmonic proper for result nonexistence a prove we note this In h , a scle a called is map A . eaReana aiodad( and manifold Riemannian a be ) 2 iamncmaps biharmonic ( steplbc ftemti tensor metric the of pullback the is ,N M, map ) k τ ( φ τ ) 2 k ( L φ φ p h nrydniyof density energy the , := ) ( M E ) > p , amncmap harmonic 2 − ( eseta amncmp r iamncmp n even and maps biharmonic are maps harmonic that see We . imninmanifolds Riemannian φ ∆ okrBadn,Yn Luo Yong Branding, Volker sgvnby given is ) E φ n mlns of smallness and 1 τ e E ( φ ( ( E 2 φ φ is ) ( ( = ) ) φ φ − = ) uy1,2018 16, July = ) τ X i | Abstract =1 m dφ ( φ 1 2 1 2 ,h N, Tr = ) | R Z 2 h Z if 1 M N h nryfntoa ftemap the of functional energy The . M Tr = φ τ ( imninmnfl ihu boundary. without manifold Riemannian a ) | ( τ e τ sdfie by defined is φ ( m ( ( φ g k φ 0. = ) φ g ∇ dφ ) ¯ dim = ( ) ) dv φ dφ , | d 2 k ∗ e h td fbhroi as The maps. biharmonic of study the ted φ dv g L h) . m ( and 0 = g e , ( M r,se[9 o survey. a for [19] see try, M i ocnie h inryfunctional bienergy the consider to )) ) 1]drvdtefis n second and first the derived [10] hsi nipoeetof improvement an is This . ≥ dφ it ftebeeg functional bienergy the of oints m < p < ihifiievlm that volume infinite with 3 ( e nvrosfilso differential of fields various in mrin rmcomplete from bmersions i 0 = ) τ z h oino harmonic of notion the ize ( φ scle the called is ) p rmcomplete from aps flsb assuming by ifolds , sapplications As . φ sdefined is tension ral φ m where ∆ := (∇¯ ∇¯ − ∇¯ ∇ ). Here, ∇ is the Levi-Civita connection on (M, g), ∇¯ is the i=1 ei ei ei ei induced connectionP on the pullback bundle φ∗TN, and RN is the Riemannian curvature tensor on N. The first nonexistence result for biharmonic maps was obtained by Jiang [10]. He proved that biharmonic maps from a compact, orientable into a Riemannian manifold of nonpositive curvature are harmonic. Jiang’s theorem is a direct application of the Weitzenb¨ock formula. If φ is biharmonic, then 1 − ∆|τ(φ)|2 = h−∆φτ(φ),τ(φ)i − |∇¯ τ(φ)|2 2 N 2 = T rghR (τ(φ), dφ)dφ, τ(φ)i − |∇¯ τ(φ)| ≤ 0.

The maximum principle implies that |τ(φ)|2 is constant. Therefore ∇¯ τ(φ)=0 and so by

divhdφ, τ(φ)i = |τ(φ)|2 + hdφ, ∇¯ τ(φ)i, we deduce that divhdφ, τ(φ)i = |τ(φ)|2. Then, by the divergence theorem, we have τ(φ) = 0. Generalizations of this result by making use of similar ideas are given in [17]. If M is non-compact, the maximum principle is no longer applicable. In this case, Baird et al. [2] proved that biharmonic maps from a complete non-compact Riemannian manifold with nonnegative into a nonpositively curved manifold with finite bienergy are harmonic. It is natural to ask whether we can abandon the curvature restriction on the domain manifold and weaken the integrability condition on the bienergy. In this direction, Nakauchi et al. [16] proved that biharmonic maps from a complete manifold to a nonpositively curved manifold are harmonic if (p = 2) 2 p (i) M |dφ| dvg < ∞ and M |τ(φ)| dvg < ∞, or p (ii)RV ol(M, g)= ∞ and RM |τ(φ)| dvg < ∞. Later Maeta [14] generalizedR this result by assuming that p ≥ 2 and further generalizations are given by the second named author in [11], [12]. Recently, the first named author proved a nonexistence result for proper biharmonic maps from complete non-compact manifolds into general target manifolds [1], by only assuming that the sectional curvatures of the target manifold have an upper bound. Explicitly, he proved the following theorem.

Theorem 1.1 (Branding). Suppose that (M, g) is a complete non-compact Riemannian mani- fold of dimension m = dim M ≥ 3 whose Ricci curvature is bounded from below and with positive injectivity radius. Let φ : (M, g) → (N, h) be a smooth biharmonic map, where N is another Riemannian manifold. Assume that the sectional curvatures of N satisfy KN ≤ A, where A is a positive constant. If p |τ(φ)| dvg < ∞ ZM and m |dφ| dvg <ǫ ZM for 2 0 (depending on p, A and the geometry of M) sufficiently small, then φ must be harmonic.

2 The central idea in the proof of Theorem 1.1 is the use of an Euclidean type Sobolev inequality that allows to control the curvature term in the biharmonic map equation. However, in order for this inequality to hold one has to make stronger assumptions on the domain manifold M as in Theorem 1.1, which we will correct below. We say that a complete non-compact Riemannian manifold of infinite volume admits an Euclidean type Sobolev inequality if the following inequality holds (assuming m = dim M ≥ 3)

− m−2 2m/(m 2) m M 2 ( |u| dvg) ≤ Csob |∇u| dvg (1.1) ZM ZM

1,2 M for all u ∈ W (M) with compact support, where Csob is a positive constant that depends on the geometry of M. Such an inequality holds in Rm and is well-known as Gagliardo-Nirenberg inequality in this case. One way of ensuring that (1.1) holds is the following: If (M, g) is a complete, non-compact Riemannian manifold of dimension m with nonnegative Ricci curvature, and if for some point x ∈ M

volg(BR(x)) lim m > 0 R→∞ ωmR m holds, then (1.1) holds true, see [18]. Here, ωm denotes the volume of the unit ball in R . For further geometric conditions ensuring that (1.1) holds we refer to [8, Section 3.7]. In this article we will correct the assumptions that are needed for Theorem 1.1 to hold and extend it to the case of p = 2, which is a more natural integrability condition. Motivated by these aspects, we actually can prove the following result:

Theorem 1.2. Suppose that (M, g) is a complete, connected non-compact Riemannian manifold of dimension m = dim M ≥ 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 curvatures satisfy KN ≤ A, where A is a positive constant. Let φ : (M, g) → (N, h) be a smooth biharmonic map. If

p |τ(φ)| dvg < ∞ ZM and m |dφ| dvg <ǫ ZM for p> 1 and ǫ> 0 (depending on p, A and the geometry of M) sufficiently small, then φ must be harmonic.

Similar ideas have been used to derive Liouville type results for p-harmonic maps in [15], see also [21] for a more general result. In the proof we choose a test function of the form − 2 p 2 (|τ(φ)| + δ) 2 τ(φ), (p> 1,δ > 0) to avoid problems that may be caused by the zero points of τ(φ). When we take the limit δ → 0, we also need to be careful about the set of zero points of τ(φ), and a delicate analysis is given. For details please see the proof in section 2. Moreover, we can get the following Liouville type result.

Theorem 1.3. Suppose that (M, g) is a complete, connected non-compact Riemannian manifold of m = dim M ≥ 3 with nonnegative Ricci curvature that admits an Euclidean type Sobolev

3 inequality of the form (1.1). Moreover, suppose that (N, h) is another Riemannian manifold whose sectional curvatures satisfy KN ≤ A, where A is a positive constant. Let φ : (M, g) → (N, h) be a smooth biharmonic map. If

p |τ(φ)| dvg < ∞ ZM and m |dφ| dvg <ǫ ZM for p> 1 and ǫ> 0 (depending on p, A and the geometry of M) sufficiently small, then φ is a constant map. Note that due to a classical result of Calabi and Yau [20, Theorem 7] a complete non-compact Riemannian manifold with nonnegative Ricci curvature has infinite volume. Remark 1.4. Due to Theorem 1.2 we only need to prove that harmonic maps satisfying the assumption of Theorem 1.3 are constant maps. Such a result was proven in [15] and thus Theorem 1.3 is a corollary of Theorem 1.5 in [15]. Conversely, Theorem 1.3 generalizes related Liouville type results for harmonic maps in [15].

Organization: Theorem 1.2 is proved in section 2. In section 3 we apply Theorem 1.2 to get several nonexistence results for proper biharmonic submersions.

2 Proof of the main result

In this section we will prove Theorem 1.2. Assume that x0 ∈ M. We choose a cutoff function 0 ≤ η ≤ 1 on M that satisfies

η(x) = 1, ∀ x ∈ BR(x0),  η(x) = 0, ∀ x ∈ M \ B2R(x0), (2.1)  C |∇η(x)| ≤ R , ∀ x ∈ M. Lemma 2.1. Let φ : (M, g)→ (N, h) be a smooth biharmonic map and assume that the sectional curvatures of N satisfy KN ≤ A. Let δ be a positive constant. Then the following inequalities hold. (1) If 1

− p − 1 2 2 p 4 2 2 (1 − ) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM 2 2 p 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |dφ| dv + (|τ(φ)| + δ) 2 dv Z g 2 Z g M R B2R(x0) − 2 2 p 4 2 2 − (p − 2) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg; (2.2) ZM

(2) If p ≥ 2, we have

− 1 2 2 p 4 2 2 η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM 2 2 p 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |dφ| dv + (|τ(φ)| + δ) 2 dv . (2.3) Z g 2 Z g M R B2R(x0)

4 − 2 2 p 2 Proof. Multiplying the biharmonic map equation by η (|τ(φ)| + δ) 2 τ(φ) we get

m − − 2 2 p 2 φ 2 2 p 2 N η (|τ(φ)| + δ) 2 h∆ τ(φ),τ(φ)i = −η (|τ(φ)| + δ) 2 R (τ(φ), dφ(ei),τ(φ), dφ(ei)). Xi=1 Integrating over M and using integration by parts we get

− 2 2 p 2 φ η (|τ(φ)| + δ) 2 h∆ τ(φ),τ(φ)idvg ZM − 2 p 2 = −2 (|τ(φ)| + δ) 2 h∇¯ τ(φ),τ(φ)iη∇ηdvg ZM − 2 2 2 p 4 − (p − 2) η |h∇¯ τ(φ),τ(φ)i| (|τ(φ)| + δ) 2 dvg ZM − 2 2 p 2 2 − η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| dvg ZM − 2 p 2 ≤ −2 (|τ(φ)| + δ) 2 h∇¯ τ(φ),τ(φ)iη∇ηdvg (2.4) ZM − 2 2 2 p 4 − (p − 2) η |h∇¯ τ(φ),τ(φ)i| (|τ(φ)| + δ) 2 dvg ZM − 2 2 p 4 2 2 − η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg. ZM Therefore when 1

− 2 2 p 2 φ η (|τ(φ)| + δ) 2 h∆ τ(φ),τ(φ)idvg ZM − p − 1 2 2 p 4 2 2 ≤ ( − 1) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM 2 2 p 2 + (|τ(φ)| + δ) 2 |∇η| dvg p − 1 ZM − 2 2 2 p 4 − (p − 2) η |h∇¯ τ(φ),τ(φ)i| (|τ(φ)| + δ) 2 dvg ZM − p − 1 2 2 p 4 2 2 ≤ ( − 1) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM C 2 p + (|τ(φ)| + δ) 2 dv R2 Z g B2R(x0) − 2 2 2 p 4 − (p − 2) η |h∇¯ τ(φ),τ(φ)i| (|τ(φ)| + δ) 2 dvg ZM − p − 1 2 2 p 4 2 2 ≤ ( − 1) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM C 2 p + (|τ(φ)| + δ) 2 dv R2 Z g B2R(x0) − 2 2 2 2 p 4 − (p − 2) η |∇¯ τ(φ)| |τ(φ)| (|τ(φ)| + δ) 2 dvg, ZM

5 where in the last inequality we used 1

|h∇¯ τ(φ),τ(φ)i|2 ≤ |∇¯ τ(φ)|2|τ(φ)|2.

Therefore we find

− p − 1 2 2 p 4 2 2 (1 − ) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM m − 2 2 p 2 N ≤ η (|τ(φ)| + δ) 2 R (τ(φ), dφ(e ),τ(φ), dφ(e ))dv Z i i g M Xi=1 C 2 p + (|τ(φ)| + δ) 2 dv R2 Z g B2R(x0) − 2 2 p 4 2 2 − (p − 2) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg ZM − 2 2 p 2 2 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |τ(φ)| |dφ| dv + (|τ(φ)| + δ) 2 dv Z g R2 Z g M B2R(x0) − 2 2 p 4 2 2 − (p − 2) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg ZM 2 2 p 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |dφ| dv + (|τ(φ)| + δ) 2 dv Z g R2 Z g M B2R(x0) − 2 2 p 4 2 2 − (p − 2) η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg, ZM which proves the first claim.

When p ≥ 2 equation (2.4) gives

− 2 2 p 2 φ η (|τ(φ)| + δ) 2 h∆ τ(φ),τ(φ)idvg ZM − 2 p 2 ≤ −2 (|τ(φ)| + δ) 2 h∇¯ τ(φ),τ(φ)iη∇ηdvg ZM − 2 2 p 4 2 2 − η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg ZM − 1 2 2 p 4 2 2 2 p 2 ≤ − η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg + 2 (|τ(φ)| + δ) 2 |∇η| dvg. 2 ZM ZM Therefore we have

− 1 2 2 p 4 2 2 η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dvg 2 ZM m p−2 C p 2 2 2 N 2 2 ≤ η (|τ(φ)| + δ) R (τ(φ), dφ(ei),τ(φ), dφ(ei))dvg + 2 (|τ(φ)| + δ) dvg ZM R ZB Xi=1 2R(x0) − 2 2 p 2 2 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |τ(φ)| |dφ| dv + (|τ(φ)| + δ) 2 dv Z g R2 Z g M B2R(x0) 2 2 p 2 C 2 p ≤ A η (|τ(φ)| + δ) 2 |dφ| dv + (|τ(φ)| + δ) 2 dv . Z g R2 Z g M B2R(x0)

6 This completes the proof of Lemma 2.1. ✷

In the following we will estimate the term

2 2 p 2 A η (|τ(φ)| + δ) 2 |dφ| dvg. ZM Lemma 2.2. Assume that (M, g) satisfies the assumptions of Theorem 1.2. Then the following inequality holds

2 2 p 2 η (|τ(φ)| + δ) 2 |dφ| dvg ZM m 2 ≤ C( |dφ| dvg) m × (2.5) ZM − 1 2 p 2 2 p 4 2 2 ( (|τ(φ)| + δ) 2 dv + η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dv ), R2 Z g Z g B2R(x0) M where C is a constant depending on p, A and the geometry of M.

2 p Proof. Set f = (|τ(φ)| + δ) 4 , then we have

2 2 p 2 2 2 2 η (|τ(φ)| + δ) 2 |dφ| dvg = η f |dφ| dvg. ZM ZM Then by H¨older’s inequality we get

2m m−2 2 2 2 2 − m η f |dφ| dvg ≤ ( (ηf) m 2 dvg) m ( |dφ| dvg) m . ZM ZM ZM Applying (1.1) to u = ηf we get

2m m−2 m−2 m M 2 ( (ηf) dvg) ≤ Csob |d(ηf)| dvg, ZM ZM which leads to

2 2 2 M m 2 2 2 2 2 η f |dφ| dvg ≤ 2Csob( |dφ| dvg) m ( |dη| f dvg + η |df| dvg). (2.6) ZM ZM ZM ZM

2 p Note that f = (|τ(φ)| + δ) 4 and

2 − 2 − 2 p 2 p 4 2 p 2 p 4 2 2 |df| = (|τ(φ)| + δ) 2 |h∇¯ τ(φ),τ(φ)i| ≤ (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| . 4 4 This completes the proof of Lemma 2.2. ✷

When 1

− p − 1 2 2 p 4 2 2 C 2 p η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dv ≤ (|τ(φ)| + δ) 2 dv , (2.7) 4 Z g R2 Z g M B2R(x0)

7 where C is a constant depending on p, A and the geometry of M. Now, set M1 := {x ∈ M|τ(φ)(x) = 0}, and M2 = M \ M1. If M2 is an empty set, then we are done. Hence we assume that M2 is nonempty and we will get a contradiction below. Note that since φ is smooth, M2 is an open set. From (2.7) we have

− p − 1 2 2 p 4 2 2 C 2 p η (|τ(φ)| + δ) 2 |∇¯ τ(φ)| |τ(φ)| dv ≤ (|τ(φ)| + δ) 2 dv . (2.8) 4 Z g R2 Z g M2 B2R(x0) Letting δ → 0 we get p − 1 C C η2|τ(φ)|p−2|∇¯ τ(φ)|2dv ≤ |τ(φ)|pdv ≤ |τ(φ)|pdv . 4 Z g R2 Z g R2 Z g M2 B2R(x0) M Letting R →∞ we get p − 1 |τ(φ)|p−2|∇¯ τ(φ)|2dv = 0. Z g 4 M2 When p ≥ 2 by a similar discussion we can prove that 1 |τ(φ)|p−2|∇¯ τ(φ)|2dv = 0. Z g 4 M2

Therefore we have that ∇¯ τ(φ) = 0 everywhere in M2 and hence M2 is an open and closed nonempty set, thus M2 = M (as we assume that M is a connected manifold) and |τ(φ)|≡ c for p some constant c 6= 0. Thus V ol(M) < ∞ by M c dvg < ∞. In the following we will need Gaffney’s theoremR [7], stated below: Theorem 2.3 (Gaffney). Let (M, g) be a complete Riemannian manifold. If a C1 1-form ω 1 satisfies that M |ω|dvg < ∞ and M |δω|dvg < ∞, or equivalently, a C vector field X defined by ω(Y )= hX,YR i, (∀Y ∈ T M) satisfiesR that M |X|dvg < ∞ and M |divX|dvg < ∞, then R R

δωdvg = divXdvg = 0. ZM ZM Define a l-form on M by

ω(X) := hdφ(X),τ(φ)i, (X ∈ T M).

Then m 2 1 |ω|dv = ( |ω(e )| ) 2 dv Z g Z i g M M Xi=1

≤ |τ(φ)||dφ|dvg ZM 1− 1 m 1 ≤ cV ol(M) m ( |dφ| dvg) m ZM < ∞.

8 m In addition, we calculate −δω = i=1(∇ei ω)(ei): P m

−δω = ∇ei (ω(ei)) − ω(∇ei ei) Xi=1 m ¯ = {h∇ei dφ(ei),τ(φ)i − hdφ(∇ei ei),τ(φ)i} Xi=1 m ¯ = h∇ei dφ(ei) − dφ(∇ei ei),τ(φ)i Xi=1 = |τ(φ)|2, where in the second equality we used ∇¯ τ(φ) = 0. Therefore

2 |δω|dvg = c V ol(M) < ∞. ZM Now by Gaffney’s theorem and the above equality we have that

2 2 0= (−δω)dvg = |τ(φ)| dvg = c V ol(M), ZM ZM which implies that c = 0, a contradiction. Therefore we must have M1 = M, i.e. φ is a . This completes the proof of Theorem 1.2. ✷

3 Applications to biharmonic submersions

In this section we give some applications of our result to biharmonic submersions. First we recall some definitions [3]. Assume that φ : (M, g) → (N, h) is a smooth map between Riemannian manifolds and x ∈ M. Then φ is called horizontally weakly conformal if either (i) dφx = 0, or ⊥ (ii) dφx maps the horizontal space Hx = {Ker dφx} conformally onto Tφ(x)N, i.e.

2 h(dφx(X), dφx(Y )) = λ g(X,Y ), (X,Y ∈Hx), for some λ = λ(x) > 0, called the dilation of φ at x. A map φ is called horizontally weakly conformal or semiconformal on M if it is horizontally weakly conformal at every point of M. Furthermore, if φ has no critical points, then we call it a horizontally conformal submersion: In this case the dilation λ : M → (0, ∞) is a smooth function. Note that if φ : (M, g) → (N, h) is a horizontally weakly conformal map and dim M < dim N, then φ is a constant map. If for every harmonic function f : V → R defined on an open subset V of N with φ−1(V ) nonempty, the composition f ◦ φ is harmonic on φ−1(V ), then φ is called a harmonic mor- phism. Harmonic morphisms are characterized as follows (cf. [6, 9]). Theorem 3.1 ([6, 9]). A smooth map φ : (M, g) → (N, h) between Riemannian manifolds is a harmonic morphism if and only if φ is both harmonic and horizontally weakly conformal.

9 When φ : (M m, g) → (N n, h), (m>n ≥ 2) is a horizontally conformal submersion, the tension field is given by

n − 2 2 1 τ(φ)= λ dφ(gradH( )) − (m − n)dφ(Hˆ ), (3.1) 2 λ2 1 1 ˆ where gradH( λ2 ) is the horizontal component of grad( λ2 ), and H is the of the fibres given by the trace m 1 Hˆ = H(∇ e ). m − n ei i i=Xn+1

Here, {ei, i = 1, ..., m} is a local orthonormal frame field on M such that {ei, i = 1, ..., n} belongs to Hx and {ej, j = n + 1, ..., m} belongs to Vx at each point x ∈ M, where TxM = Hx ⊕Vx. Nakauchi et al. [16], Maeta [14] and Luo [12] applied their nonexistence result for biharmonic maps to get conditions for which biharmonic submersions are harmonic morphisms. Here, we give another such result by using Theorem 1.2. We have Proposition 3.2. Let φ : (M m, g) → (N n, h), (m >n ≥ 2) be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold (M, g) with infinite volume, that admits an Euclidean Sobolev type inequality of the form (1.1), into a Riemannian manifold (N, h) with sectional curvatures KN ≤ A and p a real constant satisfying 1 0 (depending on p, A and the geometry of M), then φ is a harmonic morphism.

Proof. By (3.1) we have,

p p n − 2 2 1 p H ˆ |τ(φ)|hdvg = λ | λ grad ( 2 ) − (m − n)H|gdvg < ∞, ZM ZM 2 λ and since |dφ(x)|2 = nλ2(x), we get that φ is harmonic by Theorem 1.2. Since φ is also a horizontally conformal submersion, φ is a harmonic morphism by Theorem 3.1. ✷

In particular, if dim N = 2, we have Corollary 3.3. Let φ : (M m, g) → (N 2, h) be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold (M, g) with infinite volume, that admits an Euclidean Sobolev type inequality of the form (1.1), into a Riemannian surface (N, h) with Gauss curvature bounded from above and p a real constant satisfying 1

p ˆ p λ |H|gdvg < ∞, ZM and m λ dvg <ǫ ZM for sufficiently small ǫ > 0 (depending on p, A and the geometry of M), then φ is a harmonic morphism.

10 Acknowledgements: The first named author gratefully acknowledges the support of the Austrian Science Fund (FWF) through the project P30749-N35 “Geometric variational problems from string theory”. The second named author is supported by the NSF of China(No.11501421, No.11771339). Part of the work was finished when the second named author was a visiting scholar at Tsinghua University. He would like to express his gratitude to Professor Yuxiang Li and Professor Hui Ma for their invitation and to Tsinghua University for their hospitality. The second named author also would like to thank Professor Ye Lin Ou for his interest in this work and discussion.

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Volker Branding Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria [email protected]

Yong Luo School of mathematics and statistics, Wuhan university, Wuhan 430072, China [email protected]

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