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

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Some Remarks on Bi-F-Harmonic Maps and F-Biharmonic Maps SOME REMARKS ON BI-f-HARMONIC MAPS AND f-BIHARMONIC MAPS YONG LUO∗ AND YE-LIN OU ∗∗ Abstract In this paper, we prove that the class of bi-f-harmonic maps and that of f- biharmonic maps from a conformal manifold of dimension = 2 are the same (The- 6 orem 1.1). We also give several results on nonexistence of proper bi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifolds into non- positively 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). 1. A relationship between f-biharmonic maps and bi-f-harmonic maps We 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 functional 1 E(φ)= dφ 2v 2 | | g ZΩ for maps φ :(M,g) (N, h) between Riemannian manifolds and for all compact −→ domain Ω M. Harmonic map equation is simply the Euler Lagrange equation ⊆ arXiv:1808.02186v1 [math.DG] 7 Aug 2018 of the energy functional which is given (see [ES]) by (1) τ(φ) Tr dφ =0, ≡ g∇ Date: 08/06/2018. 1991 Mathematics Subject Classification. 58E20, 53C43. Key words and phrases. Biharmonic, f-biharmonic, and bi-f-harmonic maps, f-Laplacian, f-bi-Laplacian, and bi-f-Laplacian. ∗ Yong Luo was supported by the NSF of China (No.11501421, No.11771339). ∗∗ Ye-Lin Ou was supported by a grant from the Simons Foundation (#427231, Ye-Lin Ou). 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. Biharmonic map 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 Riemannian manifold −→ 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 .
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