arXiv:1808.02186v1 [math.DG] 7 Aug 2018 oiieycre imninmnfls hs nld:aybi- any include: These manifolds. Riemannian curved positively asand maps rmacmatmnfl noannpstvl uvdmnfl is manifold curved non-positively a into manifold compact a from Term16,adany and 1.6), (Theorem vrwee(hoes22ad3.3). and 2.2 (Theorems everywhere bounded n as r mohuls hyaesae otherwise. stated are they maps unless Harmonic smooth are maps, and imninmnfl noamnfl fsrcl eaiecrauehas curvature negative strictly of manifold a into manifold Riemannian oanΩ domain o maps for fteeeg ucinlwihi ie se[S)by [ES]) (see given (1) is which functional energy the of rm11.W logv eea eut nnnxsec fpoe b proper of nonexistence on results several give also We dimension 1.1). of manifold orem conformal a from maps biharmonic f b-alca,adbi- and -bi-Laplacian, 1. easm l bet tde nti ae,icuigmnfls tens manifolds, including paper, this in studied objects all assume We nti ae,w rv httecaso bi- of class the that prove we paper, this In e od n phrases. and words Key 1991 Date ∗∗ ∗ ogLowsspotdb h S fCia(o1512,No.1177 (No.11501421, China of NSF the by supported was Luo Yong eLnO a upre yagatfo h iosFudto (#4 Foundation Simons the from grant a by supported was Ou Ye-Lin eainhpbetween relationship A 08/06/2018. : ahmtc ujc Classification. Subject Mathematics OERMRSO BI- ON REMARKS SOME φ f ⊆ f ( : n bounded and bhroi asfo opeeReana aiod nonon- into manifolds Riemannian complete from maps -biharmonic M ,g M, . r rtclpit fteeeg functional energy the of points critical are amncmpequation map Harmonic ) −→ f -Laplacian. Biharmonic, ( ,h N, OGLUO YONG f f f BHROI MAPS -BIHARMONIC bhroi rsetvl,bi- (respectively, -biharmonic bery(epciey bi- (respectively, -bienrgy ewe imninmnflsadfralcompact all for and manifolds Riemannian between ) τ E ( ( φ φ ) f = ) Abstract f ≡ bhroi asadbi- and maps -biharmonic ∗ bhroi,adbi- and -biharmonic, Tr N ELNOU YE-LIN AND 82,53C43. 58E20, 1 2 maps f 1 g Z ∇ HROI ASAND MAPS -HARMONIC Ω ssml h ue arneequation Lagrange Euler the simply is dφ | d φ 0 = | 2 f v hroi asadta of that and maps -harmonic g , f f ∗∗ eeg)fo complete a from -energy) hroi maps, -harmonic f r h ae(The- same the are 2 6= hroi)mpwith map -harmonic) 1339). f 73,Y-i Ou). Ye-Lin 27231, f hroi map -harmonic -harmonic i- f f f -harmonic -harmonic -Laplacian, o fields, ion rank ≤ f 1 - 2 YONGLUO∗ AND YE-LIN OU ∗∗ where τ(φ) = Tr dφ is the tension field of the map φ. g∇ Biharmonic maps are critical points of the bienergy functional defined by 1 E (φ)= τ(φ) 2v , 2 2 | | g ZΩ where Ω is a compact domain of M and τ(φ) is the tension field of the map φ. equation (see [Ji]) is the Euler-Lagrange equation of the bienergy functional, which can be written as (2) τ (φ) := Tr [( φ φ φ )τ(φ) RN (dφ, τ(φ))dφ]=0, 2 g ∇ ∇ −∇∇M − where RN denotes the curvature operator of (N, h) defined by RN (X,Y )Z = [ N , N ]Z N Z. ∇X ∇Y −∇[X,Y ] Clearly, any harmonic map is always a biharmonic map.

f-Harmonic maps (see [Li]) are critical points of the f-energy functional defined by 1 E (φ)= f dφ 2v , f 2 | | g ZΩ where Ω is a compact domain of M and f is a positive function on M. f-Harmonic map equation can be written ( see, e.g., [Co], [OND]) as (3) τ (φ) fτ(φ) + dφ(grad f)=0, f ≡ where τ(φ) = Tr dφ is the tension field of φ. It is easily seen that an f-harmonic g∇ map with f = C is nothing but a harmonic map.

f-Biharmonic maps are critical points of the f-bienergy functional 1 E (φ)= f τ(φ) 2v , f,2 2 | | g ZΩ for maps φ : (M,g) (N, h) between Riemannian manifolds and all compact −→ domain Ω M. f-Biharmonic map equation can be written as (see [Lu]) ⊆ (4) τ (φ) fτ (φ)+(∆f)τ(φ)+2 φ τ(φ)=0, f,2 ≡ 2 ∇grad f where τ(φ) and τ2(φ) are the tension and the bitension fields of φ respectively.

Bi-f-harmonic maps φ : (M,g) (N, h) between −→ which are critical points of the bi-f-energy functional 1 (5) E (φ)= τ (φ) 2v , 2,f 2 | f | g ZΩ BI-f-HARMONIC MAPS & f-BIHARMONICMAPS 3 over all compact domain Ω of M. The Euler-Lagrange equation gives the bi-f- harmonic map equation ( [OND])

(6) τ (φ) fJ φ(τ (φ)) + φ τ (φ)=0, 2,f ≡ − f ∇grad f f φ where τf (φ) is the f-tension field of the map φ and J is the Jacobi operator of the map defined by J φ(X)= Tr [ φ φX φ X RN (dφ, X)dφ]. − g ∇ ∇ −∇∇M − We would like to point out that in some literature, e.g., [OND], [Ch], the name “f-biharmonic maps” was also used for the critical points of the bi-f-energy functional (5). From the above definitions, we can easily see the following inclusion relation- ships among the different types of biharmonic maps which give two different paths of generalizations of harmonic maps.

Harmonic maps Biharmonic maps f Biharmonic maps , { }⊂{ }⊂{ − } Harmonic maps f harmonic maps Bi f harmonic maps . { }⊂{ − }⊂{ − − } For the obvious reason, we will call a bi-f-harmonic map which is not f- harmonic a proper bi-f-harmonic map, and an f-biharmonic map which is not harmonic a proper f-biharmonic map.

Concerning harmonicity and f-harmonicity, we have a classical result of Lich- nerowicz [Li] which states that a map φ : (M m,g) (N n, h) with m = 2 is −→ 6 an f-harmonic map if and only if it is a harmonic map with respect to a metric conformal to g. Concerning the relationship between biharmonic maps and f- biharmonic maps, it was proved in [Ou2] that a map φ :(M 2,g) (N n, h) from −→ a 2-dimensional manifold is an f-biharmonic map if and only if it is a biharmonic map with respect to the metricg ¯ = f −1g conformal to g. Our first theorem shows that although f-biharmonic maps and bi-f-harmonic maps are quite different by their definitions, they belong to the same class in the sense that any bi-f-harmonic map φ : (M m,g) (N n, h) with m = 2 and −→ 6 f = α is an f-biharmonic map for f = αm/(m−2) with respect to some metric conformal to g, and conversely, any f-biharmonic map φ : (M m,g) (N n, h) −→ with m = 2 and f = α is a bi-f-harmonic map for f = α(m−2)/m with respect to 6 some metric conformal to g.

Theorem 1.1. For m =2 and a map φ :(M m,g) (N n, h) between Riemann- 6 −→ ian manifolds, we have

m (7) τ (φ,g)= f m−2 τ m (φ, g¯), 2,f f m−2 ,2 4 YONGLUO∗ AND YE-LIN OU ∗∗

2 where g¯ = f m−2 g. In particular, a map φ : (M m,g) (N n, h) is a bi- m −→ f-harmonic map if and only if it is an f m−2 -biharmonic after the conformal 2 change of the metric g¯ = f m−2 g in the domain manifold, conversely, a map −2 m n m φ : (M ,g) (N , h) is an f-biharmonic map if and only if it is a bi-f m - 2 −→ − harmonic after the conformal change of the metric g¯ = f m g.

Proof. A straightforward computation gives the transformation of the tension fields under the conformal change of a metricg ¯ = F −2g:

τ(φ, g¯) = F 2 τ(φ,g) (m 2)dφ(grad lnF ) . { − − } 2 When m = 2 and F −2 = f m−2 , we have 6 −2 −m τ(φ, g¯) = f m−2 τ(φ,g)+ f m−2 dφ(grad f) −m −m −2 −2 = f m (fτ(φ,g) + dφ(grad f)) = f m τf (φ,g).

It follows that

m −2 (8) τf (φ,g)= f m τ(φ, g¯).

Now we use formula (3) in [Ou1] to have the following formula of changes of 2 Jacobi operators under the conformal change of metricsg ¯ = f m−2 g

2 φ J (X)= f m−2 J (X)+ f −1 (X). g g¯ ∇gradf By using this and (6), we have

τ (φ,g) = fJ φ(τ (φ)) + φ τ (φ) 2,f − f ∇grad f f 2 φ φ = f f m−2 J (τ (φ))+ f −1 τ (φ + τ (φ) − g¯ f ∇gradf f ∇grad f f m (9) = f m−2 J (τ (φ)).  − g¯ f

Substituting (8) into (9) yields

m m (10) τ (φ,g) = f m−2 J f m−2 τ(φ, g¯) . 2,f − g¯  

By a straightforward computation using (see (7) in [Ou1])

J(fX)= fJ(X) (∆f)X 2 φ X, − − ∇gradf BI-f-HARMONIC MAPS & f-BIHARMONICMAPS 5 we can rewrite (10) as

m m τ (φ) = f m−2 J f m−2 τ(φ, g¯) 2,f − g¯ m  m  m −2 −2 −2 φ = f m f m J (τ(φ, g¯)+(∆ f m )τ(φ, g¯)+2 m τ(φ, g¯) g¯ g¯ m−2 − ∇gradg¯f m   = f m−2 τ m (φ, g¯), f m−2 ,2 where the third equality was obtained by using the f-bi-tension field (4) and the fact that J (τ(φ, g¯)= τ (φ, g¯). Thus, we obtain the relationship (7) from which − g¯ 2 the last statement of the theorem follows. 

Many examples of proper f-biharmonic maps were given in [Ou2], from which and Theorem 1.1 we have

Example 1. It was proved in [Ou2] that for m 3, the map φ : Rm 0 Rm x ≥ \{4 } −→ 0 with φ(x) = |x|2 is a proper f-biharmonic map for f(x) = x . Using \{ } m −8/m m| | Theorem 1.1 we conclude that the map φ :(R 0 , x δij) R 0 with x \{ } | | 4(m−2)/m−→ \{ } φ(x) = |x|2 is a proper bi-f-harmonic map for f(x) = x . In particular, 4 | | −2 4 when m = 4, we have a bi-f-harmonic map φ : (R 0 , x δij) R 0 x 2 \{ } | | −→ \{ } with φ(x)= 2 for f(x)= x . |x| | | Corollary 1.2. A map φ : (M m,g) (N n, h) with m = 2 is a bi-f-harmonic −→ 6 map if and only if

m m −2 −2 φ (11) f m τ (φ, g¯)+(∆ f m )τ(φ, g¯)+2 m τ(φ, g¯)=0, 2 g¯ m−2 ∇gradg¯f 2 −2 where g¯ = f m g, and τ(φ, g¯) and τ2(φ, g¯) are the tension and the bi-tension fields of the map φ :(M m, g¯) (N n, h) respectively. −→ m Proof. This follows from Theorem 1.1 and the f m−2 -biharmonic map equation (4). 

Proposition 1.3. A function u :(M m,g) R is bi-f-harmonic if and only if −→ it is a solution of the following bi-f-Laplace equation

2 (12) ∆f u = ∆f (∆f u)=0, where ∆f is the f-Laplace operator defined by (13) ∆ u = f∆ u + g( f, u). f ∇ ∇ Proof. For a real-valued function u : (M m,g) R, one can easily check that ∂ −→ ∂ the tension field is τ(u) = (∆u) ∂t , and hence τf (u) = f(∆u) ∂t + du(gradf) = 6 YONGLUO∗ AND YE-LIN OU ∗∗

∂ (∆f u) ∂t . It follows that the bi-f-tension field of u is given by τ (u) = fJ u(τ (u)) u τ (u) 2,f f −∇grad f f u u u N u = fTr [ τ (u) M τ (u) R (du, τ (u))du] τ (u) − g ∇ ∇ f −∇∇ f − f −∇grad f f u u u u = fTr [ τ (u) M τ (u)] τ (u) − g ∇ ∇ f −∇∇ f −∇grad f f m ∂ = f [e (e (∆ u)) M e (∆ u)] + grad f, (∆ u) − i i f −∇ei i f h ∇ f i ∂t i=1 ! X ∂ ∂ (14) = [∆ (∆ u)] = (∆2 u) , − f f ∂t − f ∂t from which the proposition follows. 

Following the practice of calling ∆2 u := ∆(∆ u) the bi-Laplacian on (M m,g) we make the following definitions.

Definition 1.4. For a function u : (M m,g) R on a Riemannian manifold, −→ we define the bi-f-Laplacian ∆2,f acting on functions by

2 (15) ∆2,f u := ∆f u = ∆f (∆f u), and the f-bi-Laplacian by

(16) ∆ u = ∆(f∆u)= f∆2 u + (∆ f)∆ u +2g( f, ∆u). f,2 ∇ ∇ 2 The solutions of the bi-f-Laplace equation ∆f u =0 and that of the f-bi-Laplace equation ∆f,2 u = 0 are called bi-f-harmonic functions and f-biharmonic func- tions respectively.

Proposition 1.5. For a real-valued function u on a Riemannian manifold (M m,g) m with m =2, we have the following relationship between bi-f-Laplacian and f m−2 - 6 bi-Laplacian operators

2 m m m (17) ∆ = ∆ u = f m−2 ∆¯ m u = f m−2 ∆ f m−2 )∆ u , 2,f f f m−2 ,2 g¯ g¯

m m  m  where ∆¯ m u := f m−2 ∆2u + (∆ f m−2 )∆ u + 2¯g(grad f m−2 , grad ∆2u) is the f m−2 ,2 g¯ g¯ g¯ g¯ g¯ g¯ m 2 f m−2 -bi-Laplacian of the conformal metric g¯ = f m−2 g. In particular, a function m u : (M m,g) R is bi-f-harmonic if and only if it is f m−2 -biharmonic with −→ 2 respect the conformal metric g¯ = f m−2 g.

Proof. Using ∆¯ and ¯ to denote the Laplacian and the operators on 2 m m −2 ∇ −2 (M , g¯ = f m g), we can easily check that ∆f u = f m ∆¯ u. A straightforward BI-f-HARMONIC MAPS & f-BIHARMONICMAPS 7 computation yields

m m 2 m−2 ¯ m−2 ¯ ∆f u = ∆f (∆f u)= f ∆(f ∆ u) m m m m = f m−2 f m−2 ∆¯ 2 u +(∆¯ f m−2 )∆¯ u +2¯g( ¯ f m−2 , ¯ ∆¯ 2 u) ∇ ∇ m (18) = f m−2 ∆¯ m u,  f m−2 ,2 from which we obtain the proposition. 

In a recent paper [Ch], Chiang proved that any bi-f-harmonic map from a com- pact manifold without boundary into a non-positively curved manifold satisfying (19) f φ φ τ (φ) φ f φ τ (φ), τ (φ) 0 h ∇ei ∇ei f −∇ei ∇ei f f i ≥ is an f-harmonic map. Our next theorem gives an improvement of this result by dropping the condition (19), and hence gives a generalization of Jiang’s result (Proposition 7 in [Ji]) on biharmonic maps viewed as bi-f-harmonic maps with f being a constant.

Theorem 1.6. Any bi-f-harmonic map φ :(M m,g) (N n, h) from a compact −→ Riemannian manifold without boundary into a non-positively curved manifold is an f-harmonic map.

Proof. A straightforward computation yields 1 ∆ τ (φ) 2 = φτ (φ) 2 + ∆φτ (φ), τ (φ) . 2 | f | |∇ f | h f f i Using this and Equation (6) we have 1 ∆ τ (φ) 2 = φτ (φ) 2 + RN (dφ,τ (φ))dφ,τ (φ) f −1 φ τ (φ), τ (φ) 2 | f | |∇ f | h f f i − h∇∇f f f i 1 = φτ (φ) 2 RN (dφ,τ (φ),dφ,τ (φ)) τ (φ) 2, ln f . |∇ f | − f f − 2h∇| f | ∇ i It follows that 1 1 ∆ τ (φ) 2 + τ (φ) 2, ln f 2 | f | 2h∇| f | ∇ i (20) = φτ (φ) 2 RN (dφ,τ (φ),dφ,τ (φ)) 0, |∇ f | − f f ≥ which implies that ∆ τ (φ) 2 0, f | f | ≥ where ∆f is the f-Laplacian defined by (13). Applying the maximum principle we conclude that τ (φ) is constant and φτ (φ)=0on M. | f | ∇ f 8 YONGLUO∗ AND YE-LIN OU ∗∗

To complete the proof, we define a vector field on M by Y = f φ, τ (φ) . h ∇ f i Then divY = τ (φ) 2 + f φ, φτ (φ) = τ (φ) 2, | f | h ∇ ∇ f i | f | which, together with Stokes’s theorem, implies that τ (φ) = 0, i.e. φ is an | f | f-harmonic map. Thus, we obtain the theorem.  From Theorem 1.6 we have Corollary 1.7. Any bi-f-harmonic map φ : (M m,g) Rn from a compact −→ manifold into a Euclidean space is a constant map. From our Theorem 1.6 and Theorem 1.1 in [HLZ] we see that any bi-f-harmonic or f-biharminic map from a closed Riemannian manifold into a Riemannian man- ifold of non-positive is an f-harmonic map or harmonic map respectively. Both these generalize Jiang’s nonexistence result which states that biharmonic maps from closed Riemannian manifolds into Riemannian manifolds of non-positive sectional curvature are harmonic maps (see Proposition 7 in [Ji]). When M is complete noncompact, nonexistence results of proper biharmonic maps into Riemannian manifolds of non-positive sectional curvature were first proved in [BFO], [NUG] and later generalized in [Ma] and [Luo1], [Luo2]. In the rest of this paper, we will give some nonexistence results of proper bi-f-harmonic maps and f-biharmonic maps which generalize the corresponding results for bi- harmonic maps obtained in [BFO], [NUG], [Ma], [Luo1], and [Luo2].

2. Some nonexistence theorems for proper bi-f-harmonic maps In this section, we give some nonexistence results of proper bi-f-harmonic maps from complete noncompact manifolds into a non-positively curved manifold. Theorem 2.1. Let φ : (M,g) (N, h) be a bi-f-harmonic map from a com- −→ plete Riemannian manifold into a Riemannian manifold of non-positive sectional curvature. Then, we have 1 (i) If ( f dφ qdv ) q < + and τ (φ) pfdv < for some q [1, ] M | | g ∞ M | f | g ∞ ∈ ∞ and p (1, ), then φ is an f-harmonic map. ∈ R ∞ R (ii) If V ol (M,g) := fdv = and τ (φ) pfdv < for some p f M g ∞ M | f | g ∞ ∈ (1, ), then φ is an f-harmonic map. ∞ R R When the target manifold has strictly negative sectional curvature, we have Theorem 2.2. Let φ :(M,g) (N, h) be a bi-f-harmonic map from a complete −→ Riemannian manifold into a Riemannian manifold of strictly negative sectional curvature with τ (φ) pfdv < for some p (1, ). If there is some point M | f | g ∞ ∈ ∞ x M such that rankφ(x) 2, then φ is an f-harmonic map. ∈ R ≥ BI-f-HARMONIC MAPS & f-BIHARMONICMAPS 9

To prove these theorems, we will need the following lemma.

Lemma 2.3. Let φ :(M,g) (N, h) be a bi-f-harmonic map from a complete −→ noncompact Riemannian manifold into a Riemannian manifold of non-positive sectional curvature. If τ (φ) pfdv < + for some p> 1, then φτ (φ)=0. M | f | g ∞ ∇ f Proof. For a real numberR ǫ> 0, a straightforward computation shows that

1 ∆( τ φ) 2 + ǫ) 2 | f | 3 1 1 (21) = ( τ (φ) 2 + ǫ)− 2 ( τ (φ) 2 + ǫ)∆ τ (φ) 2 τ (φ) 2 2 . | f | 2 | f | | f | − 4|∇| f | |   Since τ (φ) 2 =2h( φτ (φ), τ (φ)), we have ∇| f | ∇ f f τ (φ) 2 2 4( τ (φ) 2 + ǫ) φτ (φ) 2, |∇| f | | ≤ | f | |∇ f | from which we obtain 1 1 ( τ (φ) 2 + ǫ)∆ τ (φ) 2 τ (φ) 2 2 2 | f | | f | − 4|∇| f | | 1 (22) ( τ (φ) 2 + ǫ) ∆ τ (φ) 2 2 φτ (φ) 2 . ≥ 2 | f | | f | − |∇ f | Since φ is bi-f-harmonic, we use (6) to have  1 ∆ τ (φ) 2 2 | f | = φτ (φ) 2 + ∆φτ (φ), τ (φ) |∇ f | h f f i = φτ (φ) 2 Tr RN (τ (φ),dφ,τ (φ),dφ) f −1 φ τ (φ), τ (φ) |∇ f | − g f f − h∇∇f f f i 1 (23) φτ (φ) 2 τ (φ) 2, ln f , ≥ |∇ f | − 2h∇| f | ∇ i where the inequality was obtained by using the assumption that RN 0. Rewrit- ≤ ing (23) as (24) ∆ τ (φ) 2 2 φτ (φ) 2 τ (φ) 2, ln f | f | − |∇ f | ≥ −h∇| f | ∇ i and substituting (24) into the right hand side of (22) we have 1 1 ( τ (φ) 2 + ǫ)∆ τ (φ) 2 τ (φ) 2 2 2 | f | | f | − 4|∇| f | | 1 (25) ( τ (φ) 2 + ǫ) τ (φ) 2, ln f . ≥ −2 | f | h∇| f | ∇ i Using (21) and (25) we deduce that

1 1 1 ∆( τ (φ) 2 + ǫ) 2 ( τ (φ) 2 + ǫ)− 2 τ (φ) 2, ln f . | f | ≥ −2 | f | h∇| f | ∇ i 10 YONGLUO∗ AND YE-LIN OU ∗∗

Now taking the limit on both sides of the above inequality as ǫ 0, we have that → (26) ∆ τ (φ) + τ (φ) , ln f 0. | f | h∇| f | ∇ i ≥ This, by using the notation ∆ := ∆ α, , where α(x) is a function on M α −h∇ ∇·i as in [WX], can be written as (27) ∆ τ (φ) 0, − ln f | f | ≥ which means that τ (φ) is a positive ( ln f)- on M. By | f | − Theorem 4.3 in [WX], we see that if τ (φ) pfdv < for some p > 1, then M | f | g ∞ τ (φ) is a constant on M. Moreover from (23) we have φτ (φ) = 0. This | f | R ∇ f completes the proof of the lemma.  Proof of Theorem 2.1. From the above lemma we conclude that τ (φ) = c | f | is a constant. It follows that if V ol (M) = , then we must have c = 0, this f ∞ proves (ii) of Theorem 2.1. To prove (i) of Theorem 2.1, we consider two cases. If c = 0, then we are done. If c = 0, then the hypothesis implies that V ol (M) < 6 f ∞ and we will derive a contradiction as follows. Define a l-form on M by ω(X) := fdφ(X), τ (φ) , (X T M). h f i ∈ Then we have m 1 ω dv = ( ω(e ) 2) 2 dv | | g | i | g M M i=1 Z Z X f τ (φ) dφ dv ≤ | f || | g ZM 1 1 1− q q q cV olf (M) ( f dφ dvg) ≤ M | | < , Z ∞ 1 ∞ q q where if q = , we denote dφ L (M) =( M f dφ dvg) . ∞ k mk | | Now, we compute δω = ( ω)(e ) to have − i=1 ∇ei Ri m P δω = (ω(e )) ω( e ) − ∇ei i − ∇ei i i=1 Xm = φ (fdφ(e )), τ (φ) fdφ( e ), τ (φ) {h∇ei i f i−h ∇ei i f i} i=1 Xm = f φ dφ(e ) fdφ( e )+ f, φ , τ (φ) h ∇ei i − ∇ei i h∇ei ∇ei i f i i=1 X (28) = τ (φ) 2, | f | BI-f-HARMONIC MAPS & f-BIHARMONIC MAPS 11 where in obtaining the second equality we have used φτ (φ) = 0. Now by Yau’s ∇ f generalized Gaffney’s theorem (see Appendix) and (28), we have

0= δωdv = τ (φ) 2dv = c2V ol(M), − g | f | g ZM ZM which implies that c = 0, a contradiction. Therefore we must have c = 0, i.e. φ is an f-harmonic map. This completes the proof of theorem. Proof of Theorem 2.2. By Lemma 2.3, we know that τ (φ) = c, a con- | f | stant. We only need to prove that c = 0. Assume that c = 0, we will derive a 6 contradiction. It follows from (6) that at x M, we have ∈ 1 0 = ∆ τ (φ) 2 −2 | f | φ φ 2 = ∆ τf (φ), τf (φ) τf (φ) −hm i−|∇ | = RN (τ (φ),dφ(e ))dφ(e ), τ (φ) + f −1 φ τ (φ), τ (φ) φτ (φ) 2 h f i i f i h∇∇f f f i−|∇ f | i=1 Xm = RN (τ (φ),dφ(e ))dφ(e ), τ (φ) , h f i i f i i=1 X where in obtaining the first and fourth equalities we have used Lemma 2.3. Since the sectional curvature of N is strictly negative, we conclude that dφ(ei) with i =1, 2, , m are parallel to τ (φ) at any x M and hence rankφ(x) 1. This ··· f ∈ ≤ contradicts the assumption that rankφ(x) 2 for some x. The contradiction ≥ shows that we must have c = 0. Thus, we obtain the theorem. From Theorem 2.1, we obtain the following corollary. Corollary 2.4. Let φ :(M,g) (N, h) be a bi-f-harmonic map with a bounded −→ f from a complete Riemannian manifold into a Riemannian manifold of non- positive sectional curvature. If (i) f dφ 2dv < + and τ (φ) 2dv < , M | | g ∞ M | f | g ∞ or R R (ii) V ol (M,g) := fdv = and τ (φ) 2dv < , f M g ∞ M | f | g ∞ then φ is f-harmonic. R R Similarly, from Theorem 2.2 we have the following corollary which shows that a proper bi-f-harmonic map with bounded f and bounded bi-f-energy from a com- plete noncompact manifold into a negatively curved manifold must have rank 1 ≤ at any point. Corollary 2.5. Let φ :(M,g) (N, h) be a bi-f-harmonic map with bounded −→ f from a complete Riemannian manifold into a Riemannian manifold of strictly 12 YONGLUO∗ AND YE-LIN OU ∗∗ negative sectional curvature. If τ (φ) 2dv < and there is a point x M M | f | g ∞ ∈ such that rankφ(x) 2, then φ is an f-harmonic map. ≥ R 3. Some nonexistence theorems for proper f-biharmonic maps In this section we give some nonexistence theorems for proper f-biharmonic maps from a complete manifold into a non-positively curved manifold. Such a study was started in [Ou2] where it was proved that there exists no proper f- biharmonic map with constant f-bienergy density from a compact Riemannian manifold into a nonpositively curved manifold. Later in [HLZ], it was shown that the condition of having constant f-bienergy density can be dropped. When the domain manifold is complete noncompact, the authors in [HLZ] also proved the following result.

Theorem 3.1 (HLZ). Let φ : (M,g) (N, h) be an f-biharmonic map from −→ a complete Riemannian manifold (M,g) into a Riemannian manifold (N, h) with non-positive curvature. If (I) dφ 2dv < , τ(φ) 2dv < , and f p τ(φ) pdv < M | | g ∞ M | | g ∞ M | | g ∞ for some p 2, or R ≥ R R (II) V ol(M,g)= and f p τ(φ) pdv < , ∞ M | | g ∞ then φ is harmonic. R Our next theorem gives a generalization of Theorem 3.1.

Theorem 3.2. An f-biharmonic map φ : (M,g) (N, h) from a complete −→ Riemannian manifold into a Riemannian manifold of non-positive curvature is a harmonic map if either 1 (i) ( dφ qdv ) q < + and f p τ(φ) pdv < for some q [1, ] and M | | g ∞ M | | g ∞ ∈ ∞ p (1, ), or ∈ R∞ R (ii) V ol(M,g) := dv = and f p τ(φ) pdv < for some q [1, ] and M g ∞ M | | g ∞ ∈ ∞ p (1, ). ∈ ∞ R R Remark 1. Clearly, our Theorem 3.2 improves Theorem 3.1 since (I) and (II) in Theorem 3.1 (in which p 2 and finite bienergy are required) implies (i) and (ii) ≥ in Theorem 3.2 respectively, but not conversely.

When the target manifold has strictly negative sectional curvature, we have

Theorem 3.3. Let φ :(M,g) (N, h) be an f-biharmonic map from a complete −→ Riemannian manifold into a Riemannian manifold of strictly negative curvature with f p τ(φ) pdv < for some p (1, ). Assume that there is a point M | | g ∞ ∈ ∞ x M such that rankφ(x) 2, then φ is a harmonic map. ∈ R ≥ BI-f-HARMONIC MAPS & f-BIHARMONIC MAPS 13

The proofs of Theorems 3.2 and 3.3 are very similar to those of Theorems 2.1 and 2.2. The main ideas and outlines are given as follows. Proof of Theorem 3.2: First, we rewrite the f-biharmonic map equation (4) as

(29) ∆φ(fτ(φ)) Trace RN (dφ,fτ(φ))dφ =0, − g

1 and use it to compute ∆(f 2 τ(u) 2 + ǫ) 2 for a constant ǫ> 0. Then, we use the | | result and an argument similar to that used in the proof of Lemma 2.3 to have

1 ∆(f 2 τ(u) 2 + ǫ) 2 0. | | ≥

By taking limit on both sides of the above inequality as ǫ 0, we have ∆(f τ(φ) ) → | | ≥ 0. Now, by Yau’s classical Lp Liouville type theorem (see [Yau]) and the assump- tion that f p τ(φ) pdv < , we conclude that f τ(φ) = c, a constant, which M | | g ∞ | | is used to deduce that φ(fτ(φ)) = 0. R ∇ It is easy to see that f τ(φ) = c together with hypothesis (ii) implies that | | c = 0, which mean τ(φ) = 0 and hence φ is a harmonic map. To prove the theorem under hypothesis (i), we assume c = 0 and define a field of l-form ω on 6 M by

ω(X) := dφ(X), fτ(φ) , (X T M). h i ∈ Then a straightforward computation shows that δω = f τ(φ) 2 which, together − | | with the assumption (i) of Theorem 3.2, implies that

ω dv < . | | g ∞ ZM Using Yau’s generalized Gaffney’s theorem, we have

0= δωdv = f τ(φ) 2dv = c2 f −1dv , g − | | g − g ZM ZM ZM which implies that c = 0, a contradiction. The contradiction shows that we must have c = 0, and hence φ is a harmonic map. Thus, we obtain the theorem. Proof of Theorem 3.3: As in the first part of the proof of Theorem 3.2, we have f τ(φ) = c, a constant. It is enough to prove that c = 0. Assume that | | 14 YONGLUO∗ AND YE-LIN OU ∗∗ c = 0, we will derive a contradiction. Using (29) we have at x M: 6 ∈ 1 0 = ∆( fτ(φ) )2 −2 | | = ∆φ(fτ(φ)), fτ(φ) φ(fτ(φ) 2 −hm i−|∇ | = RN (fτ(φ),dφ(e ))dφ(e ), fτ(φ) φfτ(φ) 2 h i i i−|∇ | i=1 Xm = RN (fτ(φ),dφ(e ))dφ(e ), fτ(φ) . h i i i i=1 X

Since the sectional curvature of N is strictly negative, we must have that dφ(ei) with i = 1, 2, , m are parallel to fτ(φ) at any x M, so rankφ(x) 1. ··· ∈ ≤ This contradicts the assumption that rankφ(x) 2 for some x M. The ≥ ∈ contradiction shows that c = 0, and hence φ is a harmonic map. This completes the proof of Theorem 3.3.

From Theorems 3.2 and 3.3 we have the following corollaries.

Corollary 3.4. Any f-biharmonic map φ : (M,g) (N, h) with a bounded f −→ from a complete Riemannian manifold (M,g) into a Riemannian manifold (N, h) with non-positive curvature is harmonic if either (i) dφ 2dv < and f τ(φ) 2dv < , M | | g ∞ M | | g ∞ or R R (ii) V ol(M,g)= and f τ(φ) 2dv < . ∞ M | | g ∞ Corollary 3.5. Any f-biharmonicR map φ : (M,g) (N, h) with a bounded f −→ and bounded f-bienergy from a complete Riemannian manifold into a Riemannian manifold with strictly negative curvature is harmonic if there exists a point x M ∈ such that rank(φ)(x) 2. ≥ To close this section, we give the following example which shows that there does exist proper f-biharmonic map from a complete manifold into a non-positively curved manifold.

Example 2. It was proved in [Ou1] (Page 141, Example 3) that the map φ : 2 y 2 3 x x (R ,g = e R (dx + dy )) R ,φ(x, y)=(R cos , R sin ,y) is a proper bihar- −→ R R monic conformal immersion of R2 into Euclidean space R3. Then by Theorem 2.3 of [Ou2] we see that φ is a proper f-biharmonic map from a complete manifold (R2,g = dx2 + dy2) into Euclidean space R3 (a non-positively curved space) with − y f = e R . BI-f-HARMONIC MAPS & f-BIHARMONIC MAPS 15

Direct computations show that dφ = ( sin x , cos x , 0)dx + (0, 0, 1)dy and − R R τ(φ) = ( 1 cos x , 1 sin x , 0). Therefore, one can easily check that in this ex- − R R R R ample we have

dφ L∞(R2) = √2 < , ||1 || ∞ 2 dφ q R2 = √2Vol(R ) q = for q [1, ), || ||L ( ) ∞ ∈ ∞ Vol(R2)= , ∞ p p 2 f τ(φ) dvg = (p> 1), and f τ(φ) dvg = . R2 | | ∞ R2 | | ∞ Z Z So, the example shows that the hypothesis “ f p τ(φ) pdv < ” in Theorem M | | g ∞ 3.2 and the hypothesis “ f is bounded, f τ(φ) 2dv < ” in Corollary 3.4 M R| | g ∞ cannot be dropped. Note also that rank(φ)(x) = 2 for any x R2 and hence R ∈ the hypothesis “ f p τ(φ) pdv < and the target manifold has strictly neg- M | | g ∞ ative sectional curvature” in Theorem 3.3 and the hypothesis “f is bounded, R f τ(φ) 2dv < and the target manifold has strictly negative sectional cur- M | | g ∞ vature” in Corollary 3.5 can not de dropped. R 4. Further nonexistence results on proper f-biharmonic and bi-f-harmonic maps It would be interesting to know how and to what extent the nature of the func- tion f would affect the existence of proper f-biharmonic or bi-f-harmonic maps from a complete manifold into a non-positively curved manifold. In particular, one would like to know whether the boundedness assumption on the function f is essential in Corollaries 2.4, 2.5, 3.4, 3.5. In this section, we will show that we can weaken the boundedness assumption on f by replacing it with (30) sup f(x)= o(r2), asr , Br →∞ or (31) sup f(x) Cr2F (r), Br ≤ where Br is a ball of radius r centered at some point on M and F (r) is a nondecreasing function such that ∞ 1 dr = for some positive constant a rF (r) ∞ a > 0. Here, we illustrate how to generalize Corollary 2.4 by a weaker assump- R tion. Recall that in the proof of Lemma 2.3, we have ∆ τ (φ) 0, then by − ln f | f | ≥ Theorem 4.3 of [WX], we conclude that if 1 (32) lim τ (φ) 2fdv < , r→∞ r2F (r) | f | g ∞ ZBr 16 YONGLUO∗ AND YE-LIN OU ∗∗ then τ (φ) is constant and furthermore φτ (φ) = 0. It is easy to see that under | f | ∇ f the assumption of (31) and τ (φ) 2dv < , we have M | f | g ∞ R 1 2 supBr f(x) 2 limr→∞ f τf (φ) dvg lim τf (φ) dvg < , r2F (r) | | ≤ r→∞ r2F (r) | | ∞ ZBr ZM which implies (32). Therefore, τ (φ) is a constant c and furthermore φτ (φ)= | f | ∇ f 0. Thus, if V ol (M) = , we must have c = 0, i.e. φ is an f-harmonic map. f ∞ This shows that under the hypothesis (ii) of Corollary 2.4 and assumption (31), φ is an f-harmonic map. To see that Corollary 2.4 holds under hypothesis (i) of Corollary 2.4 and the assumption (30), we only need to prove that c = 0 in this case. If otherwise, we see that V ol(M) < and we will derive a contradiction ∞ as follows. Define a l-form on M by

ω(X) := fdφ(X), τ (φ) , (X T M). h f i ∈ Then we have

m 1 1 1 lim ω dv = lim ( ω(e ) 2) 2 dv r→∞ r | | g r→∞ r | i | g Br Br i=1 Z Z X 1 lim f τ (φ) dφ dv ≤ r→∞ r | f || | g ZBr 1 1 1 clim ( fdv ) 2 ( f dφ 2dv ) 2 ≤ r→∞ r g | | g ZBr ZM sup f(x) 1 1 lim Br V ol(M) 2 ( f dφ 2dv ) 2 ≤ r→∞ r | | g p ZM = 0.

On the other hand, we compute δω = m ( ω)(e ) to have − i=1 ∇ei i m P δω = (ω(e )) ω( e ) − ∇ei i − ∇ei i i=1 Xm = φ (fdφ(e )), τ (φ) fdφ( e ), τ (φ) {h∇ei i f i−h ∇ei i f i} i=1 Xm = f φ dφ(e ) fdφ( e )+ f, φ , τ (φ) h ∇ei i − ∇ei i h∇ei ∇ei i f i i=1 X = τ (φ) 2, | f | BI-f-HARMONIC MAPS & f-BIHARMONIC MAPS 17 where in obtaining the second equality we have used φτ (φ) = 0. Now by Yau’s ∇ f generalized Gaffney’s theorem(see Appendix) and the above equality we have that

0= δωdv = τ (φ) 2dv = c2V ol(M), − g | f | g ZM ZM which implies that c = 0, a contradiction. Therefore we must have c = 0, i.e. φ is an f-harmonic map. Summarizing the above discussion, we have the following theorem which gives a generalization of Corollary 2.4.

Theorem 4.1. Let φ :(M,g) (N, h) be a bi-f-harmonic map from a complete −→ Riemannian manifold (M,g) into a Riemannian manifold (N, h) of non-positive sectional curvature. If (i) sup f(x) (33) lim Br =0, r→∞ r2 where Br is a geodesic ball of radius r around some point on M, and

f dφ 2dv < + and τ (φ) 2dv < ; | | g ∞ | f | g ∞ ZM ZM or (ii) (34) sup f(x) CF (r)r2 Br ≤ for a nondecreasing function F (r) such that ∞ 1 dr = for some positive a rF (r) ∞ constant a> 0, and R V ol (M,g) := fdv = and τ (φ) 2dv < , f g ∞ | f | g ∞ ZM ZM then φ is f-harmonic.

Similar arguments apply to obtain the following theorems, which give general- izations of the corresponding results in Corollaries 2.5, 3.4 and 3.5.

Theorem 4.2. Let φ : (M,g) (N, h) be a bi-f-harmonic map from a com- −→ plete Riemannian manifold (M,g) into a Riemannian manifold (N, h) of strictly negative sectional curvature. Assume that (35) sup f(x) CF (r)r2, Br ≤ 18 YONGLUO∗ AND YE-LIN OU ∗∗ where Br is a geodesic ball of radius r around some point on M and F (r) is a nondecreasing function such that ∞ 1 dr = for some positive constant a rF (r) ∞ a> 0. Then if R τ (φ) 2dv < | f | g ∞ ZM and there is some point x M such that rankφ(x) 2, φ is an f-harmonic map. ∈ ≥ Theorem 4.3. Let φ : (M,g) (N, h) be an f-biharmonic map from a com- −→ plete Riemannian manifold (M,g) into a Riemannian manifold (N, h) of non- positive curvature. If (i) sup f(x) (36) lim Br =0, r→∞ r2 where Br is a geodesic ball of radius r around some point on M,

dφ 2dv < , and f τ(φ) 2dv < ; | | g ∞ | | g ∞ ZM ZM or (ii) (37) sup f(x) CF (r)r2 Br ≤ for a nondecreasing function F (r) such that ∞ 1 dr = for some positive a rF (r) ∞ constant a> 0, R V ol(M,g)= , and f τ(φ) 2dv < , ∞ | | g ∞ ZM then φ is a harmonic map.

Theorem 4.4. Let φ : (M,g) (N, h) be an f-biharmonic map from a com- −→ plete Riemannian manifold (M,g) into a Riemannian manifold (N, h) of strictly negative curvature. Assume that (38) sup f(x) CF (r)r2, Br ≤ where Br is a geodesic ball of radius r around some point on M and F (r) is a nondecreasing function such that ∞ 1 dr = for some positive constant a rF (r) ∞ a > 0. Then if f τ(φ) 2dv < and there exists a point x M such that M | | g ∞R ∈ rank(φ)(x) 2, φ is a harmonic map. ≥ R BI-f-HARMONIC MAPS & f-BIHARMONIC MAPS 19

5. Appendix Theorem 5.1 (Yau’s generalized Gaffney’s theorem). Let (M,g) be a complete 1 Riemannian manifold. If ω is a C 1-form such that lim ω dvg = 0, or r→∞ Br | | equivalently, a C1 vector field X defined by ω(Y )= X,Y , ( Y T M) satisfying h i ∀ R∈ lim X dvg =0, where Br is a geodesic ball of radius r around some point r→∞ Br | | on M, then R

δωdvg = divXdvg =0. ZM ZM Proof. See Appedix in [Yau].

Acknowledgements: A part of the work was done when Yong Luo was a visiting scholar at Tsinghua University. He would like to express his gratitude to Professors Yuxiang Li and Hui Ma for their invitation and to Tsinghua University for the hospitality.

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School of Mathematics and Statistics, Wuhan University Wuhan 430072, China. E-mail: [email protected]

Department of Mathematics, Texas A & M University-Commerce, Commerce, TX 75429, USA. E-mail:[email protected]