If φ:(M,g)→(N,h) is a smooth map between two Riemannian manifolds, its p-energy is defined by

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

(for more details on the concept of p-harmonic maps see [1, 3, 8]). Let τ(φ) the tension field of φ given by

where ∇M is the Levi-Civita connection of (M,g), ∇φ denote the pull-back connection on φ−1TN and {ei} is an orthonormal frame on (M,g) (see [2, 7, 18]).

Then φ is p-harmonic if and only if (see [1])

Let φ:(M,g)→(N,h) be a smooth map between two Riemannian manifolds, the (p,f)-energy is defined by

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 ⊎ 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∈(−ϵ,ϵ) a smooth variation of φ to support in D⊂M. Then

where τp,f(φ) is the (p,f)-tension field of φ given by

and v=dφtdt|t=0 denotes the variation vector field of {φt}t∈(−ϵ,ϵ).

Proof. Let ϕ:M×(−ϵ,ϵ)→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−1TN) associated to the variation {φt}t∈(−ϵ,ϵ) 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 ∇ejMei=0 at x∈M for all i,j=1,...,m (m=dimM). We compute

First, note that

Substituting the last formula in (1.8), and using ∇∂∂tϕdϕ(ei,0)=∇(ei,0)ϕdϕ(∂∂t), we obtain the following equation

Let ω∈Γ(T*M) defined by

So that, the divergence of ω at x, is given by

By the equations (1.9), (1.10), we get

The Theorem 1.1 follows from (1.11), and the divergence Theorem.

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

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 gradMf∈kerdφ. 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.

Liouville type theorems for harmonic maps between complete smooth Riemannian 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

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 Riemannian manifolds and let f∈C∞(M), then

Here ⟨,⟩ denote the inner product on T*M⊗φ−1TN.

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

where |∇dφ|2 and dφ,∇φτ(φ) are given by

Since the map φ is (p,f)-harmonic, we have

Let θ1,θ2,θ3∈Γ(T*M) defined by

where X∈Γ(TM). By using the (p,f)-harmonic condition of φ, we obtain

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

By using the Bochner formula (2.1), with RicciM≥0, SectN≤0, and HessMf≤0, we have the following inequality

We set ΔfM|dφ|2=fΔM|dφ|2+(gradMf)(|dφ|2). So, the inequality (2.2) becomes

By using the following formula

and inequality (2.3), we have the following

From the Kato's inequality |∇dφ|2−|gradM|dφ||2≥0, and the last inequality, we get

Let ρ:M→ℝ be a smooth function with compact support. Multiplying the inequality (2.5) by ρ2, with ΔfM|dφ|=divM(fgradM|dφ|), we conclude that

By the Young inequality, we have

and the following inequality

for any ϵ1,ϵ2>0. Substituting (2.7) and (2.8) in (2.6) we obtain

By using the divergence Theorem, with ϵ1=1 and ϵ2=12, we deduce

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|≤2R, where x∈ M. Replacing ρ=ρR in (2.10), we obtain

Since ∫Mf|dφ|pvg<∞, when R→∞, we have

Thus, if |dφ|≠0 on M, we have |gradM|dφ||=0, i.e. |dφ| is a positive constant on M. So that

But ∫Mfvg=∞. Hence φ is constant on M.

From Theorem 2.1, we deduce:

Corollary 2.2.

([13, 14]) Let (M, g) be a complete non-compact Riemannian manifold with positive Ricci curvature, (N,h) be Riemannian manifold with non-positive sectional curvature. If Vol(M) is infinite, then any p-harmonic map of Ep(φ)<∞ is constant.

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}(−ϵ<t<ϵ), 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 ⟨,⟩ the induced Riemannian metric on T*M⊗T*M, we have

where φ*h is the pull-back of the metric h (see [2]).

Theorem 3.1.

Under the notation above we have the following

where D is a compact subset of M, and Sp,f(φ)∈T*M⊙T*M is given by

Sp,f(φ) is called the stress (p,f)-energy tensor of φ.

Proof. Follows immediately from equations (3.1).

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 dimM=p and φ is weakly conformal.

Proof. If Sp,f(φ)=0, taking the trace shows that dimM=p, then comparing with φ*h=λ2g (where λ is a smooth function on M), shows that φ is weakly conformal, with λ=|dφ|2p.

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

Proof. Let {ei} be an orthonormal frame with respect to g on M, such that ∇ejMei=0, at x∈ M for all i, j = 1, ...,m. We compute

By the definitions of gradient and τ(φ), with ∇eiφdφ(ej)=∇ejφdφ(ei) at x, we get the following

The Theorem 3.3 follows from (3.3), and the definition of τp,f(φ).

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

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 Riemannian manifold without boundary, (N,h) is a Riemannian manifold, φ:(M,g)→(N,h) a harmonic map, assume that there is a proper homothetic vector field ξ on (N,h), that is Lξh=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≠0, and f a smooth positive function on M. If φ:(M,g)→(N,h) is (p,f)-harmonic map, satisfying

Then φ is constant.

Proof. Let ρ be a smooth function with compact support on M, we set

and let {ei} be a normal orthonormal frame at x∈ M, we have

By equation (4.2), and (p,f)-harmonicity condition of φ, we get

Since ξ is a homothetic vector field with homothetic constant k, we find that

is equivalent to the following equation

By the Young's inequality, we have

for all ϵ>0. Multiplying the last inequality by f|dφ|p−2, we get

from 4.3, 4.4, we deduce the following inequality

We assume that k>0, and we set ϵ=k2. By 4.5, we have

From 4.6, and the divergence Theorem, we have

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|≤2R, from 4.7 we get

since ∫Mf|dφ|p−2|ξ∘φ|2vg<∞, when R→∞ we obtain

Consequently, |dφ|=0 that is φ is constant (if k<0, consider the homothetic vector field ξ¯=−ξ).

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 ≠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 S1 is 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

is harmonic, where m,n,l∈ℝ. One can verify by direct computations that

is (p,f)-harmonic for all p≥2, with f(x1,x2)=e−b2cos2 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.