pISSN 1225-6951 eISSN 0454-8124 c
Kyungpook Mathematical Journal
On the Generalized of p-harmonic and f -harmonic Maps
Embarka Remli and Ahmed Mohammed Cherif∗
Mascara University, Faculty of Exact Sciences, Department of Mathematics, 29000, Algeria
e-mail : [email protected] and [email protected]
Abstract. In this paper, we extend the definition of p-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized p-harmonic maps. We present some new properties for the generalized stress p-energy tensor. We also prove that every generalized p-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.
1. (p, f )-harmonic Maps
If ϕ : (M, g) → (N, h) is a smooth map between two Riemannian manifolds, its p-energy is defined by (1.1) Ep(ϕ; D) = 1 p Z D |dϕ|pvg (p ≥ 2),
where D is a compact subset of M . We say that ϕ is a p-harmonic map if it is a critical point of the p-energy functional, that is to say, if it satisfies the Euler-Lagrange equation of the functional (1.5), that is
(1.2) τp(ϕ) ≡ divM(|dϕ|p−2dϕ) = 0,
(for more details on the concept of p-harmonic maps see [1, 3, 8]). Let τ (ϕ) the tension field of ϕ given by
(1.3) τ (ϕ) = traceg∇dϕ = ∇ϕeidϕ(ei) − dϕ(∇
M eiei),
* Corresponding Author.
Received February 15, 2020; accepted August 18, 2020.
2020 Mathematics Subject Classification: 53A45, 53C20, 58E20.
Key words and phrases: p-harmonic maps, f -harmonic maps, Liouville type theorem, stress energy tensor.
where ∇M is the Levi-Civita connection of (M, g), ∇ϕdenote the pull-back connec-tion on ϕ−1T N and {e
i} is an orthonormal frame on (M, g) (see [2, 7, 18]). Then ϕ is p-harmonic if and only if (see [1])
(1.4) |dϕ|p−2τ (ϕ) + (p − 2)|dϕ|p−3dϕ(gradM
|dϕ|) = 0.
Let ϕ : (M, g) → (N, h) be a smooth map between two Riemannian manifolds, the (p, f )-energy is defined by
(1.5) Ep,f(ϕ; D) = 1 p Z D f (x)|dϕ|pvg,
where p ≥ 2, f is a smooth positive function on M , and D is a compact subset of M . The (p, f )-energy functional (1.5) includes as a special case (f = 1) the p-energy functional, and a special case (p = 2) the f -energy functional (see [4, 6, 12, 15]).
We call (p, f )-harmonic (or generalized p-harmonic) a smooth map ϕ which is a critical point of the (p, f )-energy functional for any compact domain D.
Theorem 1.1.(The first variation of the (p, f )-energy) Let ϕ : (M, g) −→ (N, h) be a smooth map between two Riemannian manifolds, and {ϕt}t∈(−,) a smooth variation of ϕ to support in D ⊂ M . Then
(1.6) d dtEp,f(ϕt; D) t=0= − Z D h(v, τp,f(ϕ))vg, where τp,f(ϕ) is the (p, f )-tension field of ϕ given by
(1.7) τp,f(ϕ) ≡ divM(f |dϕ|p−2dϕ) = f τp(ϕ) + |dϕ|p−2dϕ(gradMf ), and v = dϕt
dt
t=0 denotes the variation vector field of {ϕt}t∈(−,).
Proof. Let φ : M × (−, ) −→ N be a smooth map defined by φ(x, t) = ϕt(x), we have φ(x, 0) = ϕ(x), and the variation vector field v ∈ (ϕ−1T N ) associated to the variation {ϕt}t∈(−,)is given by v(x) = d(x,0)φ(∂t∂), for all x ∈ M . Let {ei} be an orthonormal frame with respect to g on M , such that ∇M
ejei = 0 at x ∈ M for all i, j = 1, ..., m (m = dim M ). We compute d dtEp,f(ϕt; D) t=0= 1 p Z D f (x)∂ ∂t|dϕt| p t=0vg. (1.8)
First, note that ∂ ∂t|dϕt| p= ∂ ∂t |dϕt| 2p2 = p 2 |dϕt| 2p2−1 ∂ ∂t(|dϕt| 2) = p|dϕt|p−2h(∇ φ ∂ ∂t dφ(ei, 0), dφ(ei, 0)).
Substituting the last formula in (1.8), and using ∇φ∂ ∂t dφ(ei, 0) = ∇ φ (ei,0)dφ( ∂ ∂t), we obtain the following equation
d dtEp,f(ϕt; D) t=0= Z D f |dϕ|p−2h(∇φ(e i,0)dφ( ∂ ∂t), dφ(ei, 0)) t=0vg = Z D h(∇ϕeiv, f |dϕ|p−2dϕ(ei))vg. (1.9) Let ω ∈ Γ(T∗M ) defined by ω(X) = h(v, f |dϕ|p−2dϕ(X)), ∀X ∈ Γ(T M ). So that, the divergence of ω at x, is given by
divMω = eih(v, f |dϕ|p−2dϕ(ei)) . (1.10)
By the equations (1.9), (1.10), we get
d dtEp,f(ϕt; D) t=0= Z D (divMω)vg− Z D h(v, ∇ϕe if |dϕ| p−2dϕ(e i))vg. (1.11)
The Theorem 1.1 follows from (1.11), and the divergence Theorem. 2 From Theorem 1.1, we deduce:
Theorem 1.2. Let ϕ : (M, g) → (N, h) be a smooth map between Riemannian manifolds. Then, ϕ is (p, f )-harmonic if and only if τp,f(ϕ) = 0.
Example 1.3. According to Theorem 1.2, the inversion map ϕ : Rn\{0} −→ Rn\{0}, x 7−→ x
|x|2, is (p, f )-harmonic, for all p ≥ 2, where f (x) = |x|2(p−n)
, for all x ∈ Rn\{0}. Remark 1.4. In particular, we note that every harmonic map with constant energy density 12|dϕ|2 is (p, f )-harmonic if and only if gradM
f ∈ ker dϕ. The previous example prove the following results; There is no equivalence between the p-harmonicity of smooth map ϕ : (M, g) → (N, h) and the (p, f )-harmonicity of ϕ. There are (p, f )-harmonic maps that are neither p-harmonic nor harmonic. 2. A Liouville Type Theorem for (p, f )-harmonic Maps
Liouville type theorems for harmonic maps between complete smooth Rieman-nian manifolds have been done by many authors. Liu [10] proved the Liouville-type theorem for p-harmonic maps with free boundary values. Rimoldi and Veronelli [16] also proved the Liouville-type theorem for f -harmonic maps.
The purpose of this section is to provide a proof of the Liouville type theorem for (p, f )-harmonic maps from complete noncompact Riemannian manifold (M, g) with positive Ricci curvature into a Riemannian manifold (N, h) with non-positive sectional curvature.
Theorem 2.1. Let (M, g) be a complete non-compact Riemannian manifold with positive Ricci curvature RicciM ≥ 0, and (N, h) be a Riemannian manifold with non-positive sectional curvature SectN ≤ 0. Consider an (p, f )-harmonic map ϕ : (M, g) −→ (N, h), where f ∈ C∞(M ) is a smooth positive function, and p ≥ 2. Suppose that HessMf ≤ 0, Ep,f(ϕ) < ∞, Z M f vg= ∞. Then ϕ is constant.
We will need the following lemma to prove the Theorem 2.1.
Lemma 2.2.([5, 16]) Let ϕ : (M, g) −→ (N, h) a smooth mapping between Rieman-nian manifolds and let f ∈ C∞(M ), then
dϕ, ∇ϕdϕ(gradM f ) =1 2 grad M f |dϕ|2 + dϕ, dϕ(∇MgradM f ) . Here h, i denote the inner product on T∗M ⊗ ϕ−1T N.
Proof of Theorem 2.1. We start recalling the standard Bochner formula for the smooth map ϕ. Let {ei} be a orthonormal frame on (M, g), we have
1 2∆ M |dϕ|2= |∇dϕ|2+dϕ, ∇ϕτ (ϕ) + h dϕ RicciMe i, dϕ(ei) − h RN dϕ(e
i), dϕ(ej)dϕ(ej), dϕ(ei), (2.1)
where |∇dϕ|2anddϕ, ∇ϕτ (ϕ) are given by |∇dϕ|2= h ∇dϕ(e
i, ej), ∇dϕ(ei, ej), dϕ, ∇ϕτ (ϕ) = h dϕ(e
i), ∇ϕeiτ (ϕ).
Since the map ϕ is (p, f )-harmonic, we have
f |dϕ|p−2τ (ϕ) + (p − 2)f |dϕ|p−3dϕ(gradM|dϕ|) + |dϕ|p−2dϕ(gradMf ) = 0. Let θ1, θ2, θ3∈ Γ(T∗M ) defined by
θ1(X) = h(f |dϕ|p−2dϕ(X), τ (ϕ)), θ2(X) = |dϕ|p−2h(dϕ(X), dϕ(gradMf )),
where X ∈ Γ(T M ). By using the (p, f )-harmonic condition of ϕ, we obtain divMθ1= f |dϕ|p−2dϕ, ∇ϕτ (ϕ) , divMθ2= − 1 f|dϕ| p−2 |dϕ(gradMf )|2+ |dϕ|p−2dϕ, ∇ϕdϕ(gradMf ) . Note that by the (p, f )-harmonic condition of ϕ, we have θ1+ θ2+ θ3 = 0. From the last equations, and Lemma 2.2, we find that
divMθ3= −f |dϕ|p−2dϕ, ∇ϕτ (ϕ) + 1 f|dϕ| p−2|dϕ(gradM f )|2 −1 2|dϕ| p−2 gradM f |dϕ|2 − |dϕ|p−2dϕ, dϕ(∇MgradMf ) . By using the Bochner formula (2.1), with RicciM ≥ 0, SectN ≤ 0, and HessMf ≤ 0, we have the following inequality
divMθ3≥ f |dϕ|p−2|∇dϕ|2− 1 2f |dϕ| p−2∆M|dϕ|2 −1 2|dϕ| p−2 gradM f |dϕ|2. (2.2) We set ∆M f |dϕ|
2= f ∆M|dϕ|2+ (gradMf )(|dϕ|2). So, the inequality (2.2) becomes
divMθ3≥ f |dϕ|p−2|∇dϕ|2− 1 2|dϕ| p−2 ∆Mf |dϕ| 2 . (2.3)
By using the following formula
(2.4) 1 2∆ M f |dϕ| 2= |dϕ|∆M f |dϕ| + f | grad M |dϕ||2, and inequality (2.3), we have the following
divMθ3≥ f |dϕ|p−2|∇dϕ|2− |dϕ|p−1∆Mf |dϕ| − f |dϕ|
p−2| gradM |dϕ||2.
From the Kato’s inequality |∇dϕ|2− | gradM|dϕ||2≥ 0, and the last inequality, we get
divMθ3≥ −|dϕ|p−1∆Mf |dϕ|. (2.5)
Let ρ : M −→ R be a smooth function with compact support. Multiplying the inequality (2.5) by ρ2, with ∆M
f |dϕ| = div
M f gradM|dϕ|, we conclude that
divM(ρ2θ3) − 2(p − 2)ρf |dϕ|p−3h(dϕ(gradMρ), dϕ(gradM|dϕ|)) ≥ − divM ρ2f |dϕ|p−1gradM|dϕ|
+ 2ρf |dϕ|p−1g(gradMρ, gradM|dϕ|) + (p − 1)ρ2f |dϕ|p−2| gradM|dϕ||2. (2.6)
By the Young inequality, we have −2(p − 2)ρf |dϕ|p−3h(dϕ(gradM ρ), dϕ(gradM|dϕ|)) (2.7) ≤ 1(p − 2)ρ2f |dϕ|p−2| gradM|dϕ||2+ p − 2 1 f |dϕ|p| gradMρ|2, and the following inequality
−2ρf |dϕ|p−1g(gradM ρ, gradM|dϕ|) (2.8) ≤ 2ρ2f |dϕ|p−2| gradM|dϕ||2+ 1 2 f |dϕ|p| gradMρ|2, for any 1, 2> 0. Substituting (2.7) and (2.8) in (2.6) we obtain
divM(ρ2θ3) + p − 2 1 + 1 2 f |dϕ|p| gradMρ|2 ≥ − divM ρ2f |dϕ|p−1gradM|dϕ| (2.9) + [p − 1 − 1(p − 2) − 2] ρ2f |dϕ|p−2| gradM|dϕ||2. By using the divergence Theorem, with 1= 1 and 2=12, we deduce
(2.10) p Z M f |dϕ|p| gradMρ|2vg ≥ 1 2 Z M ρ2f |dϕ|p−2| gradM|dϕ||2v g.
Choose the smooth cut-off ρ = ρR on M , i.e. ρ ≤ 1 on M , ρ = 1 on the geodesic ball B(x, R), ρ = 0 on M \B(x, 2R) and | gradMρ| ≤ R2, where x ∈ M . Replacing ρ = ρR in (2.10), we obtain 4p R2 Z B(x,2R) f |dϕ|pvg≥ 1 2 Z B(x,R) f |dϕ|p−2| gradM|dϕ||2v g. SinceR Mf |dϕ| pv g< ∞, when R → ∞, we have Z M f |dϕ|p−2| gradM|dϕ||2= 0.
Thus, if |dϕ| 6= 0 on M , we have | gradM|dϕ|| = 0, i.e. |dϕ| is a positive constant on M . So that Ep,f(ϕ) = |dϕ|p p Z M f vg < ∞.
ButRMf vg= ∞. Hence ϕ is constant on M . 2
Corollary 2.2.([13, 14]) Let (M, g) be a complete non-compact Riemannian mani-fold with positive Ricci curvature, (N, h) be Riemannian manimani-fold with non-positive sectional curvature. If Vol(M ) is infinite, then any p-harmonic map of Ep(ϕ) < ∞ is constant.
3. Stress (p, f )-energy Tensor
Let ϕ : (M, g) −→ (N, h) be a smooth map between two Riemannian manifolds, and f ∈ C∞(M ) be a smooth positif function. Consider a smooth one-parameter variation of the metric g, i.e. is a smooth family of metrics {gt}(−<t<), such that g0= g. Write δ = ∂t∂
t=0, then δg ∈ T
∗M T∗M is a symmetric 2-covariant tensor field on M .
Let h, i the induced Riemannian metric on T∗M ⊗ T∗M , we have (3.1) δ(vgt) = 1 2hg, δgivg, δ( |dϕ|p p ) = − 1 2|dϕ| p−2hϕ∗h, δgi, (p ≥ 2)
where ϕ∗h is the pull-back of the metric h (see [2]).
Theorem 3.1. Under the notation above we have the following d dtEp,f(ϕ; D) t=0= 1 2 Z D hSp,f(ϕ), δgivg,
where D is a compact subset of M , and Sp,f(ϕ) ∈ T∗M T∗M is given by Sp,f(ϕ) =
f p|dϕ|
pg − f |dϕ|p−2ϕ∗h.
Sp,f(ϕ) is called the stress (p, f )-energy tensor of ϕ.
Proof. Follows immediately from equations (3.1). 2
From Theorem 3.1, we deduce:
Theorem 3.2. A non-constant smooth map ϕ : (M, g) −→ (N, h) is extremal with respect to variations of the metric for (p, f )-energy functional if and only if dim M = p and ϕ is weakly conformal.
Proof. If Sp,f(ϕ) = 0, taking the trace shows that dim M = p, then comparing with ϕ∗h = λ2g (where λ is a smooth function on M ), shows that ϕ is weakly conformal,
with λ = |dϕ|p2. 2
Theorem 3.3. Let ϕ : (M, g) −→ (N, h) be a smooth map between Riemannian manifolds, f a smooth positive function in M , and p ≥ 2. We have
divMSp,f(ϕ) = −h(τp,f(ϕ), dϕ) + |dϕ|p
Proof. Let {ei} be an orthonormal frame with respect to g on M , such that ∇Mejei=
0, at x ∈ M for all i, j = 1, ..., m. We compute h divMSp,f(ϕ) i (ej) = ei f p|dϕ| p δij− f |dϕ|p−2h(dϕ(ei), dϕ(ej)) = 1 pei(f )|dϕ| pδ ij+ f pei(|dϕ| p)δ ij − ei(f )|dϕ|p−2h(dϕ(ei), dϕ(ej)) − f ei(|dϕ|p−2)h(dϕ(ei), dϕ(ej)) − f |dϕ|p−2h(∇ϕ eidϕ(ei), dϕ(ej)) − f |dϕ|p−2h(dϕ(e i), ∇ϕeidϕ(ej)). (3.2)
By the definitions of gradient and τ (ϕ), with ∇ϕ
eidϕ(ej) = ∇ ϕ ejdϕ(ei) at x, we get the following h divMSp,f(ϕ) i (ej) = |dϕ|p p g(grad M f, ej) − |dϕ|p−2h(dϕ(gradMf ), dϕ(ej)) − (p − 2)f |dϕ|p−3h(dϕ(gradM |dϕ|), dϕ(ej)) − f |dϕ|p−2h(τ (ϕ), dϕ(e j)). (3.3)
The Theorem 3.3 follows from (3.3), and the definition of τp,f(ϕ). 2 4. Homothetic Vector Fields and (p, f )-harmonic Maps
A vector field ξ on a Riemannian manifold (M, g) is called a homothetic if Lξg = 2kg, for some constant k, where Lξg is the Lie derivative of the metric g with respect to ξ, that is
(4.1) g(∇MXξ, Y ) + g(∇MY ξ, X) = 2kg(X, Y ), X, Y ∈ Γ(T M ). If ξ is homothetic, while k = 0 it is Killing (see [2, 9, 19]).
In the seminal work [11], where we proved that, if (M, g) is a compact Rieman-nian manifold without boundary, (N, h) is a RiemanRieman-nian manifold, ϕ : (M, g) → (N, h) a harmonic map, assume that there is a proper homothetic vector field ξ on (N, h), that isLξh = 2kh, for some constant k ∈ R∗. Then ϕ is a constant map. We obtain the following results.
Theorem 4.1. Let (M, g) be a complete orientable Riemannian manifold, (N, h) a Riemannian manifold admitting a homothetic vector field ξ with homothetic constant k 6= 0, and f a smooth positive function on M . If ϕ : (M, g) −→ (N, h) is (p, f )-harmonic map, satisfying
Z
M
Then ϕ is constant.
Proof. Let ρ be a smooth function with compact support on M , we set ω(X) = h(ξ ◦ ϕ, ρ2f |dϕ|p−2dϕ(X)), ∀X ∈ Γ(T M ), and let {ei} be a normal orthonormal frame at x ∈ M , we have
divMω = eih(ξ ◦ ϕ, ρ2f |dϕ|p−2dϕ(ei)) . (4.2)
By equation (4.2), and (p, f )-harmonicity condition of ϕ, we get
divMω = ρ2f |dϕ|p−2h(∇ϕei(ξ ◦ ϕ), dϕ(ei)) + 2ρei(ρ)f |dϕ|p−2h(ξ ◦ ϕ, dϕ(ei)). Since ξ is a homothetic vector field with homothetic constant k, we find that divMω = kρ2f |dϕ|p−2h(dϕ(ei), dϕ(ei)) + 2ρei(ρ)f |dϕ|p−2h(ξ ◦ ϕ, dϕ(ei)), is equivalent to the following equation
divMω = kρ2f |dϕ|p+ 2ρei(ρ)f |dϕ|p−2h(ξ ◦ ϕ, dϕ(ei)). (4.3)
By the Young’s inequality, we have
−2ρei(ρ)h(ξ ◦ ϕ, dϕ(ei)) ≤ ρ2|dϕ|2+ 1 ei(ρ)
2|ξ ◦ ϕ|2,
for all > 0. Multiplying the last inequality by f |dϕ|p−2, we get (4.4) −2f |dϕ|p−2ρe i(ρ)h(ξ ◦ ϕ, dϕ(ei)) ≤ f ρ2|dϕ|p+ 1 f |dϕ| p−2e i(ρ)2|ξ ◦ ϕ|2, from (4.3), (4.4), we deduce the following inequality
kρ2f |dϕ|p− divMω ≤ f ρ2|dϕ|p+1 f |dϕ|
p−2e
i(ρ)2|ξ ◦ ϕ|2. (4.5)
We assume that k > 0, and we set = k2. By (4.5), we have k 2ρ 2f |dϕ|p− divMω ≤ 2 kf |dϕ| p−2e i(ρ)2|ξ ◦ ϕ|2. (4.6)
From (4.6), and the divergence Theorem, we have
(4.7) k 2 Z M ρ2f |dϕ|pvg≤ 2 k Z M f |dϕ|p−2ei(ρ)2|ξ ◦ ϕ|2vg.
Now, consider the cut-off smooth function ρ = ρR such that 0 ≤ ρ ≤ 1 on M , ρ = 1 on the geodesic ball B(x, R), ρ = 0 on M \ B(x, 2R) and | gradMρ| ≤R2, from (4.7) we get k 2 Z B(x,R) f |dϕ|pvg≤ 8 kR2 Z B(x,2R) f |dϕ|p−2|ξ ◦ ϕ|2vg, (4.8)
sinceRMf |dϕ|p−2|ξ ◦ ϕ|2vg< ∞, when R → ∞ we obtain Z
M
f |dϕ|pvg= 0. (4.9)
Consequently, |dϕ| = 0 that is ϕ is constant (if k < 0, consider the homothetic
vector field ¯ξ = −ξ). 2
Corollary 4.2. Let (M, g) be a compact orientable Riemannian manifold without boundary, (N, h) a Riemannian manifold admitting a homothetic vector field ξ with homothetic constant k 6= 0, f a smooth positive function on M , and p ≥ 2. Then, any (p, f )-harmonic map ϕ from (M, g) to (N, h) is constant.
Example 4.3. Let T2 = S1
× S1
the Torus. We note that the circle S1is compact orientable manifold of dimension 1, and without boundary because ∂S1
= ∂(∂D2) = ∅ where D2
is the unit disk in R2. So that the product manifold S1× S1 is also compact, without boundary, orientable manifold of dimension 2. In [17], the authors proved that the non-constant map
(T2, dx21+ dx22) −→ (S2, dy12+ sin2y1dy22), (x1, x2) 7−→ (π/2, mx1+ nx2+ l) is harmonic, where m, n, l ∈ R. One can verify by direct computations that
ϕ : (T2, dx21+ dx22) −→ (S2, dy12+ sin2y1dy22), (x1, x2) 7−→ (ax1+ c1, bx2+ c2) is (p, f )-harmonic for all p ≥ 2, with f (x1, x2) = e−
b2 cos(2 ax1+2 c1)
4a2 , where a ∈ R∗
and b, c1, c2 ∈ R. Thus, the condition of existence of the homothetic vector field with non-zero constant homothetic is necessary to verify the previous Corollary.
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