The Geometry of (p,q)-Harmonic Maps

Abstract

This paper studies $(p,q)$-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the $(p,q)$-energy functional. Second, we analyze weakly conformal and horizontally conformal $(p,q)$-harmonic maps and prove Liouville results for $(p,q)$-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the $(p,q)$-$SSU$ manifold and prove that non-constant stable $(p,q)$-harmonic maps do not exist.

1. Introduction

Harmonic maps play a fundamental role in mathematics and physics, with extensive interdisciplinary impacts [1,2,3]. These critical points of the energy functional $E\left(F\right)={\int }_M{\displaystyle \frac{{\left|dF\right|}^2}{2}}d{v}_g$ generalize naturally to p-harmonic maps, which are critical points of $E_p\left(F\right)={\int }_M{\displaystyle \frac{{\left|dF\right|}^p}{p}}d{v}_g$. In particular, when $p=2$, the 2-harmonic maps correspond precisely to the classical harmonic maps. A central research theme involves establishing Liouville-type theorems for such maps [4,5,6].

In [7], Dong and Wei established monotonicity formulas using the stress–energy tensor. Their approach relies on fundamental integral formulas and employs distinct exhaustion functions to construct the required vector fields. In [8,9], Kawai and Nakauchi showed that non-constant stable $\Phi$-harmonic maps do not exist. By using the extrinsic average variational method [10,11], in [12], Han and Wei generalized the results of Kawai and Nakauchi and proved that there are no non-constant stable $\Phi$-harmonic maps from or into these manifolds. Similarly, in [13,14,15], Feng et al. introduced the notions of ${\Phi }_{\left(3\right)}$-$SSU$, ${\Phi }_S$-$SSU$, and ${\Phi }_{S,p}$-$SSU$ manifolds. They proved that any stable ${\Phi }_{\left(3\right)}$-harmonic map from or into a compact ${\Phi }_{\left(3\right)}$-$SSU$ manifold, any stable ${\Phi }_S$-harmonic map from or into a compact ${\Phi }_S$-$SSU$ manifold, and any stable ${\Phi }_{S,p}$-harmonic map from or into a compact ${\Phi }_{S,p}$-$SSU$ manifold must be constant. In [16], Cherif investigated the stability of $P\left(x\right)$-harmonic maps. By defining core concepts such as the $P\left(x\right)$-tension field, energy functional, and index form, he proved that stable $P\left(x\right)$-harmonic maps from the two-dimensional sphere to Kähler manifolds with non-positive holomorphic bisectional curvature must be holomorphic or anti-holomorphic maps, and he clarified the constant characteristic of stable $P\left(x\right)$-harmonic maps from high-dimensional spheres to Riemannian manifolds under specific conditions.

Due to the conformal properties of harmonic maps, in the field of neuroscience, the brain can be mapped onto the unit sphere. This approach helps to simplify the complex structure of the cerebral cortex and facilitates further analysis and comparison. In [17], Nakauchi studied a variational problem related to conformal maps. In [18], Takeuchi investigated p-harmonic conformal maps and established relationships between their mean curvature vectors and p-tension fields and proved two key results: $\left(i\right)$ if $dimM=p<dimN$, a conformal map F is p-harmonic if and only if $F\left(M\right)$ is a minimal submanifold of N; $\left(ii\right)$ if $dimM=p>dimN$, the fibers of p-harmonic horizontal conformal maps are minimal submanifolds of M.

It is well-known that the single-phase functional corresponds to the classical p-energy functional, such as ${\int }_M{|\nabla F|}^{p}d{v}_g$. Single-phase problems are based on uniform ellipticity, where the energy response to the gradient is consistent across the entire domain, corresponding to the stable mechanical behavior of homogeneous materials (e.g., pure metals). However, in physical modeling, the ellipticity is not necessarily uniform. In 2015, Colombo and Mingione [19] investigated the regularity of the minimizers for the double-phase functional

$${\int }_M\left({|\nabla F|}^p+a\left(x\right){|\nabla F|}^q\right)d{v}_g.$$

Double-phase problems introduce non-uniform ellipticity by modulating the energy response via a coefficient function $a\left(x\right)$, which allows the system to switch between phases (the p-phase and the $(p,q)$-phase) depending on the spatial location. This structure is particularly suitable for modeling anisotropic materials (e.g., composite materials), where abrupt changes in mechanical properties occur across different components, thereby overcoming the limitations of the single-phase models in capturing spatially inhomogeneous energy laws. The transition from single-phase to double-phase functionals essentially aims to more accurately model the physical distinction between homogeneous and anisotropic materials.

This paper studies some properties of $(p,q)$-harmonic maps F from an m-dimensional Riemannian manifold M with metric g into an n-dimensional Riemannian manifold N with metric h. The $(p,q)$-harmonic maps are critical points of the $(p,q)$-energy functional $E_{p,q}$, which is defined below. Assume that the map $F:M\to N$ is smooth, and we consider the following functional

$$\begin{array}{c}\hfill E_{p,q}\left(F\right)={\int }_M\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g,(p,q\ge 2)\end{array}$$

(1)

where $d{v}_g$ denotes the volume form on $(M,g)$. We say that F is a $(p,q)$-harmonic map if F is a critical point of $E_{p,q}$, i.e.,

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}=0\end{array}$$

(2)

for any compactly supported variation $F_t:M\to N,(-\kappa <t<\kappa )$ with $F_0=F$. That is, F satisfies the $(p,q)$-tension field equation

$$\begin{array}{c}\hfill {\tau }_{p,q}\left(F\right)=\sum _{i=1}^m\left\{{\tilde{\nabla }}_{e_i}\left({\left(\right|dF|}^{p-2}+{\left|dF\right|}^{q-2})dF\left(e_i\right)\right)-{\left(\right|dF|}^{p-2}+{\left|dF\right|}^{q-2})dF\left({\nabla }_{e_i}{e}_i\right)\right\}\end{array}$$

on M.

Next, we extend the study of harmonic maps and p-harmonic maps, and we investigate some properties of $(p,q)$-harmonic maps. We now give the following definitions.

Definition 1. 

A smooth map F is called a $(p,q)$-harmonic map provided its $(p,q)$-tension field vanishes, i.e., ${\tau }_{p,q}\left(F\right)=0$, where ${\tau }_{p,q}\left(F\right)$ is the tension field associated with the $(p,q)$-energy functional $E_{p,q}$.

The notion of the $(p,q)$-$SSU$ manifold is defined as follows.

Definition 2. 

A Riemannian manifold $M^m$ is said to be $(p,q)$-superstrongly unstable $\left(\right(p,q)$-$SSU)$ manifold if it admits an isometric immersion into some Euclidean space ${\mathbb{R}}^d$ such that at any point $P\in M^m$, relative to its second fundamental form $\mathcal{H}$, the functional

$$\begin{array}{ccc}\hfill G_{(p,q),P}& =& max\left\{\right(p-2),(q-2\left)\right\}\langle \mathcal{H}(z,z),\mathcal{H}(z,z)\rangle \hfill \\ & & +\sum _{i=1}^m\left(2〈\mathcal{H}(z,e_i),\mathcal{H}(z,e_i)〉-〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\right)\hfill \end{array}$$

is strictly negative for all unit tangent vectors $z\in T_P\left(M^m\right)$, where ${\left\{e_i\right\}}_{i=1}^m$ is a local orthonormal frame on $M^m$.

The plan of this paper is as follows. Section 1 introduces some background knowledge and defines $(p,q)$-harmonic maps and $(p,q)$-$SSU$ manifolds. Section 2 derives the variation formulas and introduces the $(p,q)$-stress–energy tensor $S_{p,q}$ associated with the $(p,q)$-energy functional $E_{p,q}$. Section 3 establishes some properties of weakly conformal and horizontally conformal $(p,q)$-harmonic maps. Section 4 obtains a Liouville-type result (cf. Theorem 6) using Jin’s method [20]. Section 6 proves that non-constant stable $(p,q)$-harmonic maps from any compact Riemannian manifold into a compact $(p,q)$-$SSU$ manifold or from a compact $(p,q)$-$SSU$ manifold into any compact Riemannian manifold do not exist (cf. Theorem 9, Theorem 10).

2. Variation Formula and (p,q)-Stress–Energy Tensor

Let ${D, }^{N}D$, and $\tilde{D}$ be the Levi-Civita connections of M, N, and the induced connection on $F^{-1}TN$ defined by ${\tilde{D}}_{Y}W{=}^N{D}_{dF\left(Y\right)}W$, respectively, where W is a section of $F^{-1}TN$ and Y is a tangent vector of M. In [21], Takeuchi defined the tensor ${\tau }_p\left(F\right)$. We define the $(p,q)$-tension field ${\tau }_{p,q}\left(F\right)$ as follows:

$$\begin{array}{ccc}\hfill {\tau }_{p,q}\left(F\right)& =& -\delta \left({\left|dF\right|}^{p-2}dF+{\left|dF\right|}^{q-2}dF\right)\hfill \\ & =& \sum _{i=1}^m\left\{{\tilde{D}}_{e_i}\left({\left(\right|dF|}^{p-2}+{\left|dF\right|}^{q-2})dF\left(e_i\right)\right)-{\left(\right|dF|}^{p-2}+{\left|dF\right|}^{q-2})dF\left(D_{e_i}{e}_i\right)\right\},\hfill \end{array}$$

which is a section of pullback bundle $F^{-1}TN$. If ${\tau }_{p,q}\left(F\right)=0$ ($(p,q)$-harmonic map), then the physical system is in equilibrium, with no additional tension, and the energy reaches a local minimum.

Using the $(p,q)$-tension field ${\tau }_{p,q}\left(F\right)$, we get the following results.

Theorem 1. 

Suppose that the map $F:M\to N$ is smooth between Riemannnian manifolds. Then we get

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}=-{\int }_{M}h(W,{\tau }_{p,q}\left(F\right))d{v}_g,\end{array}$$

(3)

where $W={\displaystyle \frac{\partial }{\partial t}}{F}_t{|}_{t=0}$ is a vector field and $d{v}_g$ is the volume form on M.

Proof. 

Let $W\in \Gamma \left(F^{-1}TN\right)$. We may get a one-parameter family of maps $F_t$ such that $F_0=F$, $W=d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right){|}_{t=0}$. The mapping $F_t$ can be considered as a map $(-\kappa ,\kappa )\times M\to N$, and we may also use D and $\tilde{D}$ to denote the Levi-Civita connection on $(-\kappa ,\kappa )\times M$ and the induced connection on $F^{-1}TN$, respectively.

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}={\int }_M{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{|d{F}_t{|}^p}{p}}+{\displaystyle \frac{|d{F}_t{|}^q}{q}}\right){|}_{t=0}d{v}_g.\end{array}$$

(4)

Next, we calculate the expression within the following integral:

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\hfill \\ & =& {\displaystyle \frac{1}{p}}{\displaystyle \frac{\partial }{\partial t}}(|d{F}_t{|}^2{)}^{p/2}+{\displaystyle \frac{1}{q}}{\displaystyle \frac{\partial }{\partial t}}\left(\right|d{F}_t{{|}^2)}^{q/2}\hfill \\ & =& \left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^{m}h({\tilde{D}}_{{\displaystyle \frac{\partial }{\partial t}}}d{F}_t\left(e_i\right),d{F}_t\left(e_i\right))\hfill \\ & =& \left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^{m}h\left({\tilde{D}}_{e_i}d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_t\left(e_i\right)\right)\hfill \\ & =& \left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^m\left[e_{i}h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_t\left(e_i\right)\right)-h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}\left(d{F}_t\left(e_i\right)\right)\right)\right],\hfill \end{array}$$

where we use ${\tilde{D}}_{{\displaystyle \frac{\partial }{\partial t}}}d{F}_t\left(e_i\right)-{\tilde{D}}_{e_i}d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right)=d{F}_t[{\displaystyle \frac{\partial }{\partial t}},e_i]=0$ for the third equality. If for any vector field Z on M, there is $g(Y_t,Z)=h(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_t\left(Z\right))$, where $Y_t$ is a compactly supported smooth vector field on M. Therefore, we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\hfill \\ & =& \left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^m\left[e_{i}g(Y_t,e_i)-h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}\left(d{F}_t\left(e_i\right)\right)\right)\right]\hfill \\ & =& \left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\mathrm{div}{Y}_t\hfill \\ & & -\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^{m}h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}\left(d{F}_t\left(e_i\right)\right)-d{F}_t\left(D_{e_i}{e}_i\right)\right)\hfill \\ & =& \mathrm{div}\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)Y_t\right)-\sum _{i=1}^m{e}_i\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\right)h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_t\left(e_i\right)\right)\hfill \\ & & -\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)\sum _{i=1}^{m}h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}\left(d{F}_t\left(e_i\right)\right)-d{F}_t\left(D_{e_i}{e}_i\right)\right)\hfill \\ & =& \mathrm{div}\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)Y_t\right)\hfill \\ & & -\sum _{i=1}^{m}h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)d{F}_t\left(e_i\right)\right)-\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)d{F}_t\left(D_{e_i}{e}_i\right)\right)\hfill \\ & =& \mathrm{div}\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)Y_t\right)-h\left(d{F}_t\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tau }_{p,q}\left(F_t\right)\right).\hfill \end{array}$$

(5)

According to Green’s theorem,

$${\int }_M\mathrm{div}\left(\left(|d{F}_t{|}^{p-2}+{\left|d{F}_t\right|}^{q-2}\right)Y_t\right)d{v}_g=0.$$

Integrating both sides of (5) and substituting into (4), we have the first variational formula

$${\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}=-{\int }_{M}h(W,{\tau }_{p,q}\left(F\right))d{v}_g.$$

□

Remark 1. 

Let $(M,g)$ be an oriented, compact, m-dimensional Riemannian manifold without boundary. Then for any smooth vector field Y,

$${\int }_M(divY)d{v}_g=0.$$

Let $(M,g)$ be an oriented, compact, m-dimensional Riemannian manifold with boundary. Let $\mathit{n}$ be the unit normal vector field on $\partial M$ pointing inward to M. Then for any smooth vector field Y,

$${\int }_M(divY)d{v}_g=-{\int }_{\partial M}g(\mathit{n},Y)d{v}_g,$$

where $\partial M$ has orientation induced by M.

Next, we define the $(p,q)$-stress–energy tensor

$$\begin{array}{ccc}\hfill S_{p,q}\left(F\right)& =& \left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)g-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)F^{\ast }h.\hfill \end{array}$$

Proposition 1. 

If the map $F:M\to N$ is smooth, for any smooth vector fields Y on M, the relation between $(p,q)$-tension field ${\tau }_{p,q}\left(F\right)$ and $(p,q)$-stress–energy tensor $S_{p,q}\left(F\right)$ can be written as

$$\begin{array}{c}\hfill \left(\mathrm{div}{S}_{p,q}\right)\left(Y\right)=-h({\tau }_{p,q}\left(F\right),dF\left(Y\right)).\end{array}$$

(6)

Proof. 

We choose a local orthonormal frame ${\left\{e_i\right\}}_{i=1}^m$ of M near a point P. Let Y be a vector field on M. At P, we have

$$\begin{array}{ccc}& & \left(\mathrm{div}{S}_{p,q}\right)\left(Y\right)=\sum _{i=1}^m\left(D_{e_i}{S}_{p,q}\right)(e_i,Y)=\sum _{i=1}^m[e_i{S}_{p,q}(e_i,Y)-S_{p,q}(D_{e_i}{e}_i,Y)-S_{p,q}(e_i,D_{e_i}Y)]\hfill \\ & =& Y\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\sum _{i=1}^m\left\{e_i\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left(Y\right),dF\left(e_i\right)\right)\right.\hfill \\ & & +\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)[h({\tilde{D}}_{e_i}dF\left(e_i\right),dF\left(Y\right))+h(dF\left(e_i\right),{\tilde{D}}_{e_i}dF\left(Y\right))]\hfill \\ & & \left.-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h(dF\left(D_{e_i}{e}_i\right),dF\left(Y\right))-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h(dF\left(e_i\right),dF\left(D_{e_i}Y\right))\right\}\hfill \\ & =& Y\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(dF\left(e_i\right),\left(D_{e_i}dF\right)\left(Y\right))-h\left({\tau }_{p,q}\left(F\right),dF\left(Y\right)\right)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m[h(\left(D_{Y}dF\right)\left(e_i\right),dF\left(e_i\right))-h(dF\left(e_i\right),\left(D_{e_i}dF\right)\left(Y\right))]\hfill \\ & & -h({\tau }_{p,q}\left(F\right),dF\left(Y\right)),\hfill \end{array}$$

since $\left(D_{e_i}dF\right)\left(Y\right)=\left(D_{Y}dF\right)\left(e_i\right)$, we have $\left(\mathrm{div}{S}_{p,q}\right)\left(Y\right)=-h({\tau }_{p,q}\left(F\right),dF\left(Y\right)).$ □

Therefore, if ${\tau }_{p,q}\left(F\right)=0$, then

$$\begin{array}{c}\hfill \left(\mathrm{div}{S}_{p,q}\right)\left(Y\right)=0,\end{array}$$

(7)

i.e., F satisfies $(p,q)$-conservation law.

Let $T_1,T_2\in \Gamma (T^{\ast }M\otimes T^{\ast }M)$ be any two 2-tensors. Their inner product is defined by

$$\begin{array}{c}\hfill \langle T_1,T_2\rangle =\sum _{i,j=1}^m{T}_1(e_i,e_j)T_2(e_i,e_j).\end{array}$$

(8)

Assume that $\Omega$ is any bounded domain of M and $\Omega$ has $C^1$ boundary. Let $\chi$ be the unit outward normal vector field along $\partial \Omega$. By Stokes’s theorem, we have

$$\begin{array}{c}\hfill {\int }_{\partial \Omega }T(Y,\chi )d{s}_g={\int }_{\Omega }\left[〈T,{\displaystyle \frac{1}{2}}{L}_{Y}g〉+\mathrm{div}\left(T\right)\left(Y\right)\right]d{v}_g.\end{array}$$

(9)

Setting $T=S_{p,q}$ in (9), and using (7), we obtain

$$\begin{array}{c}\hfill {\int }_{\partial \Omega }{S}_{p,q}(Y,\chi )d{s}_g={\int }_{\Omega }〈S_{p,q},{\displaystyle \frac{1}{2}}{L}_{Y}g〉d{v}_g.\end{array}$$

(10)

Theorem 2. 

Assume that the map $F:M\to N$ is smooth. Then we have

$${\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}=-{\int }_M〈S_{p,q},{\displaystyle \frac{1}{2}}{L}_{Y}g〉d{v}_g,$$

where $L_Y$ is the Lie derivative, and for a local orthonormal frame field $\left\{e_i\right\}$ on M, the inner product is expressed as

$$\langle S_{p,q},L_{Y}g\rangle =\sum _{i,j}{S}_{p,q}(e_i,e_j)L_{Y}g(e_i,e_j).$$

Proof. 

Let $Y\in {\Gamma }_0\left(TM\right)$, and let ${\phi }_t^Y(-\kappa <t<\kappa )$ be a one-parameter family of diffeomorphisms of M, hence $dF\left(Y\right)$ is the variation vector field for $F_t=F\circ {\phi }_t^Y$. By Theorem 1, we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}=-{\int }_M\langle dF\left(Y\right),{\tau }_{p,q}\left(F\right)\rangle d{v}_g\hfill \\ & =& -{\int }_M〈dF\left(Y\right),\sum _{i=1}^m{\tilde{\nabla }}_{e_i}\left(\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\left(e_i\right)\right)〉d{v}_g\hfill \\ & & +{\int }_M〈dF\left(Y\right),\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}dF\left({\nabla }_{e_i}{e}_i\right)〉d{v}_g\hfill \\ & =& -{\int }_M{e}_i〈dF\left(Y\right),\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\left(e_i\right)〉d{v}_g\hfill \\ & & +{\int }_M〈{\tilde{\nabla }}_{e_i}dF\left(Y\right),\left(\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\left(e_i\right)\right)〉d{v}_g\hfill \\ & & +{\int }_M〈dF\left(Y\right),\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}dF\left({\nabla }_{e_i}{e}_i\right)〉d{v}_g\hfill \\ & =& {\int }_M\mathrm{div}\left(h(dF\left(Y\right),\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\left(e_i\right))e_i\right)d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m\langle {\tilde{\nabla }}_{e_i}dF\left(Y\right),dF\left(e_i\right)\rangle d{v}_g.\hfill \end{array}$$

(11)

According to the Green theorem. Take a local orthonormal frame ${\left\{e_i\right\}}_{i=1}^m$ near a fixed point $P\in M$ such that $D_{e_i}{e}_j{|}_P=0$. At P, we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{d}{dt}}{E}_{p,q}\left(F_t\right){|}_{t=0}\hfill \\ & =& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m\langle {\tilde{D}}_{e_i}dF\left(Y\right),dF\left(e_i\right)\rangle d{v}_g.\hfill \\ & =& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m[h(\left(D_{e_i}dF\right)\left(Y\right),dF\left(e_i\right))+h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))]d{v}_g\hfill \\ & =& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m[h(\left(D_{Y}dF\right)\left(e_i\right),dF\left(e_i\right))+h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))]d{v}_g\hfill \\ & =& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m[h({\tilde{D}}_{Y}dF\left(e_i\right),dF\left(e_i\right))+h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))]d{v}_g\hfill \\ & =& {\int }_M\left[D_Y\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)+\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))\right]d{v}_g\hfill \\ & =& {\int }_M\left[L_Y\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)+\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))\right]d{v}_g\hfill \\ & =& -{\int }_M\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\mathrm{div}Yd{v}_g+{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(dF\left(D_{e_i}Y\right),dF\left(e_i\right))d{v}_g\hfill \\ & =& -{\int }_M\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\sum _{i,j=1}^{m}g(e_i,e_j)g(D_{e_i}Y,e_j)d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j=1}^{m}h(dF\left(e_j\right),dF\left(e_i\right))g(D_{e_i}Y,e_j)d{v}_g\hfill \\ & =& -{\int }_M\sum _{i,j=1}^m{S}_{p,q}(e_i,e_j)g(D_{e_i}Y,e_j)d{v}_g\hfill \\ & =& -{\int }_M〈S_{p,q},{\displaystyle \frac{1}{2}}{L}_{Y}g〉d{v}_g.\hfill \end{array}$$

□

From Theorem 2 and Equation (7), we obtain the following corollary.

Corollary 1. 

If the map $F:M\to N$ is $(p,q)$-harmonic, then we have

$$\begin{array}{c}\hfill {\int }_M〈S_{p,q},{\displaystyle \frac{1}{2}}{L}_{Y}g〉d{v}_g=0,\end{array}$$

(12)

where Y is a vector field.

Next, we derive the following second-variation formula:

Theorem 3. 

Assume that $F:M\to N$ is a $(p,q)$-harmonic map, and let $F_{s,t}:M\to N(-\kappa <s,t<\kappa )$ be a compact two-parameter variation so that $F_{0,0}=F$. Let $V={\displaystyle \frac{\partial }{\partial s}}{F}_{s,t}{|}_{s,t=0}$, $W={\displaystyle \frac{\partial }{\partial t}}{F}_{s,t}{|}_{s,t=0}$. Then we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\partial }^2}{\partial s\partial t}}{E}_{p,q}\left(F_{s,t}\right){|}_{s,t=0}\hfill \\ & =& (p-2){\int }_M{\left|dF\right|}^{p-4}\sum _{i=1}^m\langle {\tilde{D}}_{e_i}W,dF\left(e_i\right)\rangle \sum _{j=1}^m\langle {\tilde{D}}_{e_j}V,dF\left(e_j\right)\rangle d{v}_g\hfill \\ & & +(q-2){\int }_M{\left|dF\right|}^{q-4}\sum _{i=1}^m\langle {\tilde{D}}_{e_i}W,dF\left(e_i\right)\rangle \sum _{j=1}^m\langle {\tilde{D}}_{e_j}V,dF\left(e_j\right)\rangle d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^m({\tilde{D}}_{e_i}V,{\tilde{D}}_{e_i}W)d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(R^N(W,dF\left(e_i\right))V,dF\left(e_i\right))d{v}_g,\hfill \end{array}$$

(13)

where $R^N$ denotes the curvature tensor.

Ordering

$$\begin{array}{c}\hfill I(W,V)={\displaystyle \frac{{\partial }^2}{\partial s\partial t}}{E}_{p,q}\left(F_{s,t}\right){|}_{s,t=0}.\end{array}$$

(14)

If $I(W,W)\ge 0$ for all compactly supported vector fields W along F, then the $(p,q)$-harmonic map F is said to be stable.

Proof. 

Let $V,W\in \Gamma \left(F^{-1}TN\right)$ be vector fields along F. We consider a two-parameter family of maps $F_{s,t}$ such that $F_{0,0}=F$. Let $V={\displaystyle \frac{\partial }{\partial s}}{F}_{s,t}{|}_{s,t=0}$, $W={\displaystyle \frac{\partial }{\partial t}}{F}_{s,t}{|}_{s,t=0}$. The mapping $F_{s,t}$ can be considered as a map $(-\kappa ,\kappa )\times (-\kappa ,\kappa )\times M\to N$, and we denote by D the Levi-Civita connection on $(-\kappa ,\kappa )\times (-\kappa ,\kappa )\times M$, and by $\tilde{D}$ the induced connection on $F^{-1}TN$. Fix a point $P\in M$. We take a local orthonormal frame $\{e_1,\cdots ,e_m\}$ on M near P satisfying $D_{e_i}{e}_j{|}_P=0$.

Using (3), the definition of the $(p,q)$-harmonic map and $[e_i,{\displaystyle \frac{\partial }{\partial s}}]=0$, we have at P,

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\partial }^2}{\partial s\partial t}}{E}_{p,q}\left(F_{s,t}\right){|}_{s,t=0}\hfill \\ & =& -{\displaystyle \frac{\partial }{\partial s}}{\int }_{M}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tau }_{p,q}\left(F_{s,t}\right)\right)d{v}_g{|}_{s,t=0}\hfill \\ & =& -{\int }_{M}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),\sum _{i=1}^m{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}{\tilde{D}}_{e_i}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)d{v}_g{|}_{s,t=0}\hfill \\ & & +{\int }_{M}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),\sum _{i=1}^m{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(D_{e_i}{e}_i\right)\right)\right)d{v}_g{|}_{s,t=0}\hfill \\ & =& -{\int }_{M}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),\sum _{i=1}^m{\tilde{D}}_{e_i}{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)d{v}_g{|}_{s,t=0}\hfill \\ & & -{\int }_{M}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),\sum _{i=1}^m{R}^N\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right),d{F}_{s,t}\left(e_i\right)\right)\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)d{v}_g{|}_{s,t=0}.\hfill \end{array}$$

(15)

Next, we calculate the first expression on the right-hand side of Equation (15):

$$\begin{array}{ccc}\hfill & & h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),\sum _{i=1}^m{\tilde{D}}_{e_i}{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)\hfill \\ & =& \sum _{i=1}^m{e}_{i}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)\hfill \\ & & -\sum _{i=1}^{m}h\left({\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{{\displaystyle \frac{\partial }{\partial s}}}\left((|d{F}_{s,t}{|}^{p-2}+|d{F}_{s,t}{|}^{q-2})d{F}_{s,t}\left(e_i\right)\right)\right)\hfill \\ & =& \sum _{i=1}^m(p-2)e_i\left\{|d{F}_{s,t}{|}^{p-4}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_{s,t}\left(e_i\right)\right)\sum _{j=1}^{m}h\left({\tilde{D}}_{e_j}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right),d{F}_{s,t}\left(e_j\right)\right)\right\}\hfill \\ & & +(q-2)\sum _{i=1}^m{e}_i\left\{|d{F}_{s,t}{|}^{q-4}h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_{s,t}\left(e_i\right)\right)\sum _{j=1}^{m}h\left({\tilde{D}}_{e_j}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right),d{F}_{s,t}\left(e_j\right)\right)\right\}\hfill \\ & & +\sum _{i=1}^m{e}_i\left\{\left(|d{F}_{s,t}{|}^{p-2}+{|d{F}_{s,t}|}^{q-2}\right)h\left(d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right)\right)\right\}\hfill \\ & & -(p-2)|d{F}_{s,t}{|}^{p-4}\sum _{i=1}^{m}h\left({\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_{s,t}\left(e_i\right)\right)\sum _{j=1}^{m}h\left({\tilde{D}}_{e_j}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right),d{F}_{s,t}\left(e_j\right)\right)\hfill \\ & & -(q-2)|d{F}_{s,t}{|}^{q-4}\sum _{i=1}^{m}h\left({\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),d{F}_{s,t}\left(e_i\right)\right)\sum _{j=1}^{m}h\left({\tilde{D}}_{e_j}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right),d{F}_{s,t}\left(e_j\right)\right)\hfill \\ & & -\left(|d{F}_{s,t}{|}^{p-2}+{|d{F}_{s,t}|}^{q-2}\right)\sum _{i=1}^{m}h\left({\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial t}}\right),{\tilde{D}}_{e_i}d{F}_{s,t}\left({\displaystyle \frac{\partial }{\partial s}}\right)\right).\hfill \end{array}$$

(16)

For any vector field Z on M, we take compactly supported vector fields $Y_1,Y_2$, and $Y_3$ on M satisfying

$$\begin{array}{ccc}& & g(Y_1,Z)={\left|dF\right|}^{p-4}\sum _{j=1}^m\langle {\tilde{D}}_{e_j}V,dF\left(e_j\right)\rangle h(dF\left(Z\right),W),\hfill \\ & & g(Y_2,Z)={\left|dF\right|}^{q-4}h(W,dF\left(Z\right))\sum _{j=1}^{m}h({\tilde{D}}_{e_j}V,dF\left(e_j\right)),\hfill \\ & & g(Y_3,Z)=\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h(W,{\tilde{D}}_{Z}V).\hfill \end{array}$$

When $s,t=0$, according to Green’s formula, we have

$$\begin{array}{ccc}\hfill & & {\int }_M\sum _{i=1}^m\left[(p-2)e_{i}g(Y_1,e_i)+(q-2)e_{i}g(Y_2,e_i)+e_{i}g(Y_3,e_i)\right]d{v}_g\hfill \\ & =& {\int }_M\left[(p-2)\mathrm{div}\left(Y_1\right)+(q-2)\mathrm{div}\left(Y_2\right)+\mathrm{div}\left(Y_3\right)\right]d{v}_g\hfill \\ & =& 0,\hfill \end{array}$$

(17)

hence, the theorem is proved by (15)–(17). □

3. The Results of (Horizontally) Conformal (p,q)-Harmonic Maps

In this section, we get some results for (horizontally) conformal $(p,q)$-harmonic maps by the definition of a (weakly) conformal map (cf. [15]).

Definition 3. 

Let $F:(M,g)\to (N,h)$ be a smooth map, so we have the following:

$\left(i\right)$ F is said to be a conformal map if there exists a smooth positive function $\mu :(M,g)\to \mathbb{R}$ on M such that $F^{\ast }h={\mu }^{2}g$.

$\left(ii\right)$ F is said to be a weakly conformal map if there exists a smooth non-negative function $\mu :(M,g)\to \mathbb{R}$ on M such that $F^{\ast }h={\mu }^{2}g$.

Proposition 2. 

If $\left(a\right)m=p,m\ne q,\left(b\right)m\ne p,m=q,\mathrm{or}\left(c\right)m\ne p,m\ne q$ and the map $F:M\to N$ is a $(p,q)$-harmonic and weakly conformal, then F is homothetic.

Proof. 

Choose a local orthonormal frame ${\left\{e_i\right\}}_{i=1}^m$ on M such that ${\left(D{e}_j\right)}_P=0$ at $P\in M$. Since $F:M\to N$ is a weakly conformal map, there is a smooth non-negative function $\mu$ on M satisfying $F^{\ast }h={\mu }^{2}g$, so

$$\begin{array}{ccc}\hfill S_{p,q}\left(F\right)& =& \left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)g-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)F^{\ast }h\hfill \\ & =& \left({\displaystyle \frac{m^{p/2}{\mu }^p}{p}}g-m^{p/2-1}{\mu }^{p-2}{\mu }^{2}g+{\displaystyle \frac{m^{q/2}{\mu }^q}{q}}g-m^{q/2-1}{\mu }^{q-2}{\mu }^{2}g\right)\hfill \\ & =& \left[{\displaystyle \frac{m^{\frac{p}{2}-1}(m-p)}{p}}\right]{\mu }^{p}g+\left[{\displaystyle \frac{m^{\frac{q}{2}-1}(m-q)}{q}}\right]{\mu }^{q}g.\hfill \end{array}$$

(18)

Using the definition of the $(p,q)$-harmonic map and Equation (18), we have

$$\begin{array}{ccc}\hfill 0& =& \left(\mathrm{div}{S}_{p,q}\right)\left(Y\right)\hfill \\ & =& \sum _{i=1}^m\left\{D_{e_i}\left[{\displaystyle \frac{m^{\frac{p}{2}-1}(m-p)}{p}}{\mu }^{p}g\right](e_i,Y)+D_{e_i}\left[{\displaystyle \frac{m^{\frac{q}{2}-1}(m-q)}{q}}{\mu }^{q}g\right](e_i,Y)\right\}\hfill \\ & =& \sum _{i=1}^m{\displaystyle \frac{m^{\frac{p}{2}-1}(m-p)}{p}}{e}_i\left({\mu }^p\right)g(e_i,Y)+\sum _{i=1}^m{\displaystyle \frac{m^{\frac{q}{2}-1}(m-q)}{q}}{e}_i\left({\mu }^q\right)g(e_i,Y)\hfill \\ & =& {\displaystyle \frac{m^{\frac{p}{2}-1}(m-p)}{p}}Y\left({\mu }^p\right)+{\displaystyle \frac{m^{\frac{q}{2}-1}(m-q)}{q}}Y\left({\mu }^q\right),\hfill \end{array}$$

where Y is a vector field on M. Thus,

(a)

$\mathrm{If}m=p,m\ne q$, then ${\mu }^q$ is constant and F is homothetic;

(b)

$\mathrm{If}m\ne p,m=q$, then ${\mu }^p$ is constant and F is homothetic;

(c)

$\mathrm{If}m\ne p,m\ne q$, then $\mu$ is constant and F is homothetic.

□

Remark 2. 

This analog is of a harmonic map due to Eells and Lemaire [22] which states that if $m>2$ and $F:(M,g)\to (N,h)$ is harmonic and conformal, then F is homothetic.

Rigidity: Homothetic maps scale metrics uniformly. This forbids local shearing or warping—preserving shapes up to global scaling, for example, triangles map to similar triangles.

Next, suppose that the map $F:M\to N$ is smooth. For each $P\in M$ satisfying $d{F}_P\ne 0$, let $H_P$ be the orthogonal complement of $V_P=\mathrm{Ker}d{F}_P\subset T_{P}M$. Denote by $H_P$ the horizontal space and by $V_P$ the vertical space. For $Y\in T_{P}M$, $Y=Y^H+Y^V$, where $Y^H\in H_P$ and $Y^V\in V_P$. A map F is a horizontally conformal if there is positive smooth function $\mu$ on M satisfying $h(dF\left(Y\right),dF\left(Z\right))={\mu }^{2}g(Y,Z)$ for any two smooth vectors $Y,Z\in H_P$. The function $\mu$ is said to be the dilation of F.

Remark 3. 

A conformal map preserves the angles across the entire tangent space. However, a horizontally conformal map preserves angles only within the horizontal distribution; angles in directions transverse to this distribution may not be preserved. For example, consider the map

$$u:(x,y,z)\mapsto (\lambda x,\lambda y,az).\lambda >0$$

If $a=\lambda$, then this is a conformal map, as it preserves angles by uniformly scaling all directions. If $a\ne \lambda$, then the map is horizontally conformal with respect to the given horizontal distribution, since the scaling factor is the same within the horizontal directions; however, in the vertical direction, the scaling factor is not λ, which means that the angles are preserved only in the specific horizontal directions. This illustrates the characteristic property of a horizontally conformal map.

Theorem 4. 

Suppose that $F:M\to N$ is a horizontally weakly conformal $(p,q)$-harmonic map and F has dilation μ, where $m>n>max\{p,q\}$. Then the following results are equivalent:

$\left(i\right)$ The fibers of the map F are minimal submanifolds;

$\left(ii\right)$$\mathit{grad}\left({\mu }^p\right)$ and $\mathit{grad}\left({\mu }^q\right)$ are vertical;

$\left(iii\right)$ The mean curvature of the horizontal distribution is ${\displaystyle \frac{q{n}^{p/2}grad\left({\mu }^p\right)+p{n}^{q/2}grad\left({\mu }^q\right)}{pq\left(n^{p/2}{\mu }^p+n^{q/2}{\mu }^q\right)}}$.

Proof. 

We take a local orthonormal frame field ${\left\{e_i\right\}}_{i=1}^m$ near P on M, where $e_1,\cdots ,e_n$ are horizontal and $e_{n+1},\cdots ,e_m$ are vertical. Given that F is horizontally weakly conformal with dilation $\mu$, and let $Y,Z$ be smooth vector fields on M, then

$$\begin{array}{ccc}\hfill S_{p,q}(Y,Z)& =& \left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)g(Y,Z)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h(dF\left(Y\right),dF\left(Z\right))\hfill \\ & =& {\displaystyle \frac{n^{p/2}{\mu }^p}{p}}g(Y,Z)-n^{p/2-1}{\mu }^{p-2}h(dF\left(Y\right),dF\left(Z\right))\hfill \\ & & +{\displaystyle \frac{n^{q/2}{\mu }^q}{q}}g(Y,Z)-n^{q/2-1}{\mu }^{q-2}h(dF\left(Y\right),dF\left(Z\right)),\hfill \end{array}$$

(19)

Using the definition of the $(p,q)$-harmonic map and Equation (19), we have

$$\begin{array}{ccc}\hfill 0& =& \left(\mathrm{div}{S}_{p,q}\right)\left(e_j\right)=\sum _{i=1}^m\left(D_{e_i}{S}_{p,q}\right)(e_i,e_j)\hfill \\ & =& \sum _{i=1}^m\left\{e_i\left({\displaystyle \frac{n^{p/2}{\mu }^p}{p}}g(e_i,e_j)-n^{p/2-1}{\mu }^{p-2}h(dF\left(e_i\right),dF\left(e_j\right))\right)\right.\hfill \\ & & -\left({\displaystyle \frac{n^{p/2}{\mu }^p}{p}}g(D_{e_i}{e}_i,e_j)-n^{p/2-1}{\mu }^{p-2}h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))\right)\hfill \\ & & \left.-\left({\displaystyle \frac{n^{p/2}{\mu }^p}{p}}g(e_i,D_{e_i}{e}_j)-n^{p/2-1}{\mu }^{p-2}h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right)\right\}\hfill \\ & & +\sum _{i=1}^m\left\{e_i\left({\displaystyle \frac{n^{q/2}{\mu }^q}{q}}g(e_i,e_j)-n^{q/2-1}{\mu }^{q-2}h(dF\left(e_i\right),dF\left(e_j\right))\right)\right.\hfill \\ & & -\left({\displaystyle \frac{n^{q/2}{\mu }^q}{q}}g(D_{e_i}{e}_i,e_j)-n^{q/2-1}{\mu }^{q-2}h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))\right)\hfill \\ & & \left.-\left({\displaystyle \frac{n^{q/2}{\mu }^q}{q}}g(e_i,D_{e_i}{e}_j)-n^{q/2-1}{\mu }^{q-2}h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right)\right\}\hfill \\ & =& \sum _{i=1}^m\left\{{\displaystyle \frac{n^{p/2}}{p}}{e}_i\left({\mu }^p\right)g(e_i,e_j)+{\displaystyle \frac{n^{q/2}}{q}}{e}_i\left({\mu }^q\right)g(e_i,e_j)\right.\hfill \\ & & -n^{p/2-1}{e}_i\left({\mu }^{p-2}h(dF\left(e_i\right),dF\left(e_j\right))\right)-n^{q/2-1}{e}_i\left({\mu }^{q-2}h(dF\left(e_i\right),dF\left(e_j\right))\right)\hfill \\ & & +n^{p/2-1}{\mu }^{p-2}\left[h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))+h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right]\hfill \\ & & \left.+n^{q/2-1}{\mu }^{q-2}\left[h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))+h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right]\right\}.\hfill \end{array}$$

(20)

On the other hand, for $1\le j\le n$, we get

$$\begin{array}{ccc}\hfill 0& =& \sum _{i=1}^m{e}_{i}g(e_i,e_j)=\sum _{i=1}^m[g(D_{e_i}{e}_i,e_j)+g(e_i,D_{e_i}{e}_j)]\hfill \\ & =& \sum _{i=1}^m[g({\left(D_{e_i}{e}_i\right)}^H,e_j)+g(e_i,{\left(D_{e_i}{e}_j\right)}^H)]\hfill \\ & =& {\displaystyle \frac{1}{{\mu }^2}}\sum _{i=1}^n[h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))+h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))],\hfill \end{array}$$

i.e.,

$$\begin{array}{c}\hfill \sum _{i=1}^n[h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))+h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))]=0.\end{array}$$

(21)

From (20) and (21), we get

$$\begin{array}{ccc}\hfill 0& =& {\displaystyle \frac{n^{p/2}}{p}}{e}_j\left({\mu }^p\right)+{\displaystyle \frac{n^{q/2}}{q}}{e}_j\left({\mu }^q\right)-n^{p/2-1}{e}_j\left({\mu }^p\right)-n^{q/2-1}{e}_j\left({\mu }^q\right)\hfill \\ & & +\left(n^{p/2-1}{\mu }^{p-2}+n^{q/2-1}{\mu }^{q-2}\right)\sum _{i=n+1}^{m}h(dF\left(D_{e_i}{e}_i\right),dF\left(e_j\right))\hfill \\ & =& {\displaystyle \frac{n^{\frac{p}{2}-1}(n-p)}{p}}{e}_j\left({\mu }^p\right)+{\displaystyle \frac{n^{\frac{q}{2}-1}(n-q)}{q}}{e}_j\left({\mu }^q\right)\hfill \\ & & +\left(n^{p/2-1}{\mu }^p+n^{q/2-1}{\mu }^q\right)\sum _{i=n+1}^{m}g(D_{e_i}{e}_i,e_j),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill & & n^{{\displaystyle \frac{p-q}{2}}}\left[{\displaystyle \frac{n-p}{p}}{e}_j\left({\mu }^p\right)e_j+{\mu }^p\sum _{i=n+1}^{m}g(D_{e_i}{e}_i,e_j)e_j\right]\hfill \\ & & +{\displaystyle \frac{n-q}{q}}{e}_j\left({\mu }^q\right)e_j+{\mu }^q\sum _{i=n+1}^{m}g(D_{e_i}{e}_i,e_j)e_j=0.\hfill \end{array}$$

(22)

The mean curvature $\mathbf{H}$ of the fiber of F is as follows

$$\begin{array}{c}\hfill \mathbf{H}={\displaystyle \frac{1}{m-n}}\sum _{j=1}^n\sum _{i=n+1}^{m}g(D_{e_i}{e}_i,e_j)e_j.\end{array}$$

(23)

Equations (22) and (23) imply

$$\begin{array}{c}\hfill n^{{\displaystyle \frac{p-q}{2}}}\left({\displaystyle \frac{n-p}{p}}{\left(grad{\mu }^p\right)}^H+(m-n){\mathbf{H}}^{\prime }\right)+\left({\displaystyle \frac{n-q}{q}}{\left(grad{\mu }^q\right)}^H+(m-n){\mathbf{H}}^{″}\right)=0.\end{array}$$

(24)

Since the fibers of the map F are minimal submanifolds, $H^{\prime }=H^{″}=0$. From (24), ${\left(grad{\mu }^p\right)}^H={\left(grad{\mu }^q\right)}^H=0$. Thus, Equation (24) implies that $\left(i\right)⟺\left(ii\right)$.

For $n+1\le j\le m$, from (20), we get

$$\begin{array}{ccc}\hfill 0& =& {\displaystyle \frac{n^{p/2}}{p}}{e}_j\left({\mu }^p\right)+{\displaystyle \frac{n^{q/2}}{q}}{e}_j\left({\mu }^q\right)\hfill \\ & & +\left(n^{p/2-1}{\mu }^{p-2}+n^{q/2-1}{\mu }^{q-2}\right)\sum _{i=1}^{n}h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\hfill \\ & =& n^{p/2-1}\left({\displaystyle \frac{n}{p}}{e}_j\left({\mu }^p\right)+{\mu }^{p-2}\sum _{i=1}^{n}h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right)\hfill \\ & & +n^{q/2-1}\left({\displaystyle \frac{n}{q}}{e}_j\left({\mu }^q\right)+{\mu }^{q-2}\sum _{i=1}^{n}h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))\right).\hfill \end{array}$$

(25)

For $1\le i\le n$, we get

$$\begin{array}{ccc}\hfill h(dF\left(e_i\right),dF\left(D_{e_i}{e}_j\right))& =& h(dF\left(e_i\right),dF\left({\left(D_{e_i}{e}_j\right)}^H\right))={\mu }^{2}g(e_i,{\left(D_{e_i}{e}_j\right)}^H)\hfill \\ & =& {\mu }^{2}g(e_i,D_{e_i}{e}_j)=-{\mu }^{2}g(D_{e_i}{e}_i,e_j).\hfill \end{array}$$

(26)

From (25) and (26), we have

$$\begin{array}{ccc}\hfill 0& =& n^{p/2-1}\left[{\displaystyle \frac{n}{p}}{e}_j\left({\mu }^p\right)-{\mu }^p\sum _{i=1}^{n}g(D_{e_i}{e}_i,e_j)\right]\hfill \\ & & +n^{q/2-1}\left[{\displaystyle \frac{n}{q}}{e}_j\left({\mu }^q\right)-{\mu }^q\sum _{i=1}^{n}g(D_{e_i}{e}_i,e_j)\right].\hfill \end{array}$$

The mean curvature ${\mathbf{H}}^1$ of the horizontal distribution can be expressed as follows:

$$\begin{array}{ccc}\hfill {\mathbf{H}}^1& =& {\displaystyle \frac{1}{n}}\sum _{j=n+1}^m\sum _{i=1}^{n}g(D_{e_i}{e}_i,e_j)e_j\hfill \\ & =& {\displaystyle \frac{n^{p/2}}{p\left(n^{p/2}{\mu }^p+n^{q/2}{\mu }^q\right)}}\sum _{j=n+1}^m{e}_j\left({\mu }^p\right)e_j+{\displaystyle \frac{n^{q/2}}{q\left(n^{p/2}{\mu }^p+n^{q/2}{\mu }^q\right)}}\sum _{j=n+1}^m{e}_j\left({\mu }^q\right)e_j\hfill \\ & =& {\displaystyle \frac{n^{p/2}\left[grad\left({\mu }^p\right)-{\left(grad\left({\mu }^p\right)\right)}^H\right]}{p\left(n^{p/2}{\mu }^p+n^{q/2}{\mu }^q\right)}}+{\displaystyle \frac{n^{q/2}\left[grad\left({\mu }^q\right)-{\left(grad\left({\mu }^q\right)\right)}^H\right]}{q\left(n^{p/2}{\mu }^p+n^{q/2}{\mu }^q\right)}}.\hfill \end{array}$$

(27)

So $\left(ii\right)⟺\left(iii\right)$. From the above, $\left(i\right)⟺\left(ii\right)⟺\left(iii\right)$.

□

4. Liouville-Type Theorem

Suppose that M, equipped with the pole $P_0$, is complete. Let $r\left(P\right)=dis{t}_g(P,P_0)$, $B\left(r\right)=\{P\in M^m:r\left(P\right)\le r\}$. It is well-known that ${\displaystyle \frac{\partial }{\partial r}}$ is an eigenvector of $Hes{s}_g\left(r^2\right)$ with respect to eigenvalue 2. ${\lambda }_{max}$ (or ${\lambda }_{min}$) denotes the maximum (or minimum) eigenvalue of $Hes{s}_g\left(r^2\right)-2dr\otimes dr$ at any point of $M\setminus \left\{P_0\right\}$.

Theorem 5. 

Suppose that the map $F:M\to N$ is smooth between manifolds and satisfies Equation (12) for any $Y\in \Gamma \left(TM\right)$. If there is a positive constant ϱ satisfying

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{m-1}{2}}{\lambda }_{min}-{\displaystyle \frac{max\{p,q\}}{2}}max\{2,{\lambda }_{max}\}\ge \varrho ,\end{array}$$

(28)

then, for any $0<{\rho }_1\le {\rho }_2$, we have

$${\displaystyle \frac{{\int }_{B\left({\rho }_1\right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_1^{\varrho }}}\le {\displaystyle \frac{{\int }_{B\left({\rho }_2\right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_2^{\varrho }}}.$$

Proof. 

Set $Y=\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}=\zeta \left(r\right){\displaystyle \frac{1}{2}}D{r}^2$, where $\zeta \left(r\right)$ is a non-negative function. Let $\{e_1,···,e_m\}$ be an orthonormal basis associated with g and $e_m={\displaystyle \frac{\partial }{\partial r}}$. Suppose that $Hes{s}_g\left(r^2\right)$ is a diagonal matrix associated with ${\left\{e_i\right\}}_{i=1}^m$. Now let us calculate

$$\begin{array}{ccc}\hfill & & 〈S_{p,q},L_{\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}}g〉\hfill \\ & =& \sum _{i,j=1}^m{S}_{p,q}(e_i,e_j)\left(L_{\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}}g\right)(e_i,e_j)\hfill \\ & =& \sum _{i,j=1}^m\left[\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)g(e_i,e_j)\left(L_{\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}}g\right)(e_i,e_j)\right.\hfill \\ & & \left.-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h(dF\left(e_i\right),dF\left(e_j\right))\left(L_{\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}}g\right)(e_i,e_j)\right]\hfill \\ & =& \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\sum _{i=1}^{m}Hes{s}_g\left(r^2\right)(e_i,e_i)+2{\zeta }^{\prime }\left(r\right)r\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\hfill \\ & & -\zeta \left(r\right)\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j=1}^{m}h(dF\left(e_i\right),dF\left(e_j\right))Hes{s}_g\left(r^2\right)(e_i,e_j)\hfill \\ & & -2{\zeta }^{\prime }\left(r\right)r\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left({\displaystyle \frac{\partial }{\partial r}}\right),dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right)\hfill \\ & \ge & \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)[2+(m-1){\lambda }_{min}]\hfill \\ & & -\zeta \left(r\right)max\{2,{\lambda }_{max}\}\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i=1}^{m}h(dF\left(e_i\right),dF\left(e_i\right))\hfill \\ & & +2{\zeta }^{\prime }\left(r\right)r\left[\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left({\displaystyle \frac{\partial }{\partial r}}\right),dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right)\right]\hfill \\ & \ge & \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)[2+(m-1){\lambda }_{min}-max\{p,q\}·max\{2,{\lambda }_{max}\}]\hfill \\ & & +2{\zeta }^{\prime }\left(r\right)r\left[\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left({\displaystyle \frac{\partial }{\partial r}}\right),dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right)\right].\hfill \end{array}$$

(29)

By (28) and (29), we have

$$\begin{array}{ccc}\hfill & & 〈S_{p,q},{\displaystyle \frac{1}{2}}{L}_{\zeta \left(r\right)r{\displaystyle \frac{\partial }{\partial r}}}g〉\ge \varrho \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\hfill \\ & & +{\zeta }^{\prime }\left(r\right)r\left[\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left({\displaystyle \frac{\partial }{\partial r}}\right),dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right)\right].\hfill \end{array}$$

(30)

By (12) and (30), we have

$$\begin{array}{ccc}\hfill & & 0\ge {\int }_M\left\{\varrho \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\right.\hfill \\ & & \left.+{\zeta }^{\prime }\left(r\right)r\left[\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)-\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)h\left(dF\left({\displaystyle \frac{\partial }{\partial r}}\right),dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right)\right]\right\}d{v}_g.\hfill \end{array}$$

(31)

Take a positive number $\kappa$ and fix it. Let $\xi$ be a smooth function on $[0,\infty )$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\xi \left(r\right)={\xi }_{\kappa }\left(r\right)=\left\{\begin{array}{ccc}1,\hfill & & if0\le r\le 1,\hfill \\ 0,\hfill & & if1+\kappa \le r,\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(32)

and ${\displaystyle \frac{d}{dt}}\xi \left(r\right)\le 0$. Define $\zeta \left(r\right)={\zeta }_{\rho }\left(r\right)=\xi \left({\displaystyle \frac{r}{\rho }}\right)$. It is easy to see that

$$\begin{array}{c}\hfill {\zeta }^{\prime }\left(r\right)r=-\rho {\displaystyle \frac{d{\zeta }_{\rho }\left(r\right)}{d\rho }},{\zeta }^{\prime }\left(r\right)={\displaystyle \frac{1}{\rho }}{\xi }^{\prime }\left({\displaystyle \frac{r}{\rho }}\right)\le 0.\end{array}$$

(33)

By (31) and (33), we have

$$\begin{array}{ccc}\hfill 0& \ge & {\int }_M\left\{\varrho \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)+{\zeta }^{\prime }\left(r\right)r\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\right.\hfill \\ & & -\left.{\zeta }^{\prime }\left(r\right)r\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left|dF\left({\displaystyle \frac{\partial }{\partial r}}\right)\right|}^2\right\}d{v}_g\hfill \\ & \ge & {\int }_M\left\{\varrho \zeta \left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)+{\zeta }^{\prime }\left(r\right)r\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)\right\}d{v}_g\hfill \\ & =& \varrho {\int }_M{\zeta }_{\rho }\left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g-\rho {\displaystyle \frac{d}{d\rho }}{\int }_M{\zeta }_{\rho }\left(r\right)\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g,\hfill \end{array}$$

(34)

so we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\rho }}{\displaystyle \frac{{\int }_M{\zeta }_{\rho }\left(r\right)\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }^{\varrho }}}\ge 0.\end{array}$$

(35)

Hence, for any $0<{\rho }_1\le {\rho }_2$, we have

$${\displaystyle \frac{{\int }_{B\left((1+\kappa ){\rho }_1\right)}{\zeta }_{{\rho }_1}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_1^{\varrho }}}\le {\displaystyle \frac{{\int }_{B\left((1+\kappa ){\rho }_2\right)}{\zeta }_{{\rho }_2}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_2^{\varrho }}}.$$

Let $\kappa \to 0$ and note that ${\zeta }_{\rho }=1$ on $B\left(\rho \right)$, then

$${\displaystyle \frac{{\int }_{B\left({\rho }_1\right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_1^{\varrho }}}\le {\displaystyle \frac{{\int }_{B\left({\rho }_2\right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }_2^{\varrho }}}.$$

□

From [23,24,25,26,27], we have the following corollary.

Corollary 2. 

Suppose that M, equipped with a pole $P_0$, is complete. Let $K_r$ denote the radial curvature of M.

(1) 

If $-{\beta }^2\ge K_r\ge -{\alpha }^2$ with $\alpha \ge \beta \ge 0$, and if $(m-1)\beta -max\{p,q\}\alpha >0$, then

$$1+{\displaystyle \frac{m-1}{2}}{\lambda }_{min}-{\displaystyle \frac{max\{p,q\}}{2}}max\{2,{\lambda }_{max}\}\ge m-{\displaystyle \frac{max\{p,q\}\alpha }{\beta }}.$$

(2) 

If ${\displaystyle \frac{B}{{(1+r^2)}^{1+ϵ}}}\ge K_r\ge -{\displaystyle \frac{A}{{(1+r^2)}^{1+ϵ}}}$ with $ϵ>0$, $A\ge 0$, and $B\in [0,2ϵ]$, then

$$1+{\displaystyle \frac{m-1}{2}}{\lambda }_{min}-{\displaystyle \frac{max\{p,q\}}{2}}max\{2,{\lambda }_{max}\}\ge 1+(m-1)\left(1-{\displaystyle \frac{B}{2ϵ}}\right)-max\{p,q\}{e}^{{\displaystyle \frac{A}{2ϵ}}}.$$

(3) 

If ${\displaystyle \frac{b^2}{1+r^2}}\ge K_r\ge -{\displaystyle \frac{a^2}{1+r^2}}$ with $a\ge 0$ and $b^2\in \left[0,\frac{1}{4}\right]$, then

$$\begin{array}{ccc}& & 1+{\displaystyle \frac{m-1}{2}}{\lambda }_{min}-{\displaystyle \frac{max\{p,q\}}{2}}max\{2,{\lambda }_{max}\}\hfill \\ & \ge & 1+(m-1){\displaystyle \frac{1+\sqrt{1-4b^2}}{2}}-max\{p,q\}{\displaystyle \frac{1+\sqrt{1+4a^2}}{2}}.\hfill \end{array}$$

From Theorem 5 and Corollary 2, we obtain the following proposition.

Proposition 3. 

Let $F:(M,g)\to ({\mathbb{R}}^n,h)$ be a $(p,q)$-harmonic map. If $F\left(M\right)⫋S^{n-1}$ and $r\left(P\right)$ satisfies $\left(28\right)$, then as $R\to \infty$, we have

$${\int }_{B\left(R\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g\ge c\left(F\right)R^{\varrho },$$

where $c\left(F\right)>0$ is a constant determined by F.

Proof. 

Since F satisfies the condition in Theorem 5, for any $0<\rho <R$, we have

$${\displaystyle \frac{{\int }_{B\left(\rho \right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }^{\varrho }}}\le {\displaystyle \frac{{\int }_{B\left(R\right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{R^{\varrho }}}.$$

Since $F\left(M\right)⫋S^{n-1}$, there exists $\rho >0$ such that

$${\int }_{B\left(\rho \right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g>0.$$

Let

$$c\left(F\right)={\displaystyle \frac{{\int }_{B\left(\rho \right)}\left(\frac{{\left|dF\right|}^p}{p}+\frac{{\left|dF\right|}^q}{q}\right)d{v}_g}{{\rho }^{\varrho }}},$$

then

$${\int }_{B\left(R\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g\ge c\left(F\right)R^{\varrho }.$$

□

Following the approach in [6], we require F to satisfy $\left(f_1\right)$. There exists $0<\tilde{\varrho }<\varrho$ (where $\varrho$ is defined in (28)) such that

$${\left[\underset{r\left(P\right)=R}{max}{h}^2(F\left(P\right),X_0)\right]}^2\le R^{\tilde{\varrho }}{\left[{\int }_R^{\infty }{\displaystyle \frac{dr}{{\left[\mathrm{vol}(\partial B\left(r\right))\right]}^{\frac{1}{y-1}}}}\right]}^{y-1}\mathrm{for}r\left(P\right)\gg 1,$$

where $X_0\in S^{n-1}$ is a point, and $y\in \{p,q\}$. For simplicity, in Section 4, y is taken to be either p or q.

Theorem 6. 

Suppose that the $(p,q)$-harmonic map $F:(M,g)\to ({\mathbb{R}}^n,h)$ is smooth and $r\left(P\right)$ satisfies $\left(28\right)$. If $F\left(P\right)\to X_0\in S^{n-1}$ as $r\left(P\right)\to \infty$ and F satisfies $\left(f_1\right)$, then the map F is necessarily constant, or there exist $R_0>0$ such that

$$E_{p,q}^R\left(F\right)\le \left(C(p,q)R^{\tilde{\varrho }-\varrho }+{\displaystyle \frac{c\left(F\right)}{R^{\varrho }}}\right)R^{\varrho }\mathit{for}R\ge R_0.$$

Proof. 

Suppose F is non-constant, then Proposition 3 implies

$$\underset{R\to \infty }{lim}{E}_{p,q}^R\left(F\right)=\underset{R\to \infty }{lim}{\int }_{B\left(R\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g=\infty .$$

Given $X_0=({\epsilon }_1,\cdots ,{\epsilon }_n)\in S^{n-1}$ with ${\sum }_{\alpha =1}^n{\epsilon }_{\alpha }^2=1$, there exists an orthogonal matrix A such that $A{X}_0={\tilde{X}}_0=({\tilde{\epsilon }}_1,\cdots ,{\tilde{\epsilon }}_n)$ and ${\tilde{\epsilon }}_{\alpha }\ne 0$ for $\alpha =1,\cdots ,n$. If F is a $(p,q)$-harmonic map, then so is $AF$. Thus, we may assume $F\left(P\right)\to X_0=({\epsilon }_1,\cdots ,{\epsilon }_n)$ (where ${\epsilon }_{\alpha }\ne 0$ for $\alpha =1,\cdots ,n$) as $r\left(P\right)\to \infty$.

This asymptotic behavior guarantees that there exists an $R_1>0$ and a neighborhood U of $X_0$ such that for $r\left(P\right)>R_1$, $F\left(P\right)\in U$ and $F_{\alpha }\ne 0$ for $\alpha =1,\cdots ,n$. The proof has four steps.

Step 1:

Construct a perturbation map and select a specific perturbation function.

For $G\in C_0^2(M\setminus B\left(R_1\right),U)$, we define the perturbed map as follows:

$$(F+tG)\left(Q\right)=\left\{\begin{array}{cc}F\left(Q\right),\hfill & \mathrm{if}Q\in B\left(R_1\right)\hfill \\ (F+tG)\left(Q\right),\hfill & \mathrm{if}Q\in M\setminus B\left(R_1\right)\hfill \end{array}\right.$$

for sufficiently small t. Since F is a $(p,q)$-harmonic map, we have

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{E}_{p,q}(F+tG){|}_{t=0}=0.$$

Using Einstein notation, we have

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial G_{\alpha }}{\partial x_j}}d{v}_g+{\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial G_{\beta }}{\partial x_l}}d{v}_g=0.\hfill \end{array}$$

(36)

To ensure that the perturbation term G satisfies the compact support condition while preserving the functional structure, and given that the cut-off function $\eta$ localizes the global analysis problem to asymptotic regions before recovering the original global properties via a limiting process, we therefore now set $G\left(P\right)=\eta \left(r\left(P\right)\right)\tilde{F}\left(P\right)$ in (36) for $\eta \left(r\right)\in C_0^{\infty }(R_1,\infty )$, where ${\tilde{F}}_{\alpha }={\displaystyle \frac{F_{\alpha }^2-{\epsilon }_{\alpha }^2}{F_{\alpha }}}$. We get

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}\eta \left(r\left(P\right)\right)d{v}_g\hfill \\ & & +{\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial \eta \left(r\right(P\left)\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & & +{\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}\eta \left(r\left(P\right)\right)d{v}_g\hfill \\ & & +{\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial \eta \left(r\right(P\left)\right)}{\partial x_l}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\tilde{F}}_{\beta }d{v}_g=0.\hfill \end{array}$$

(37)

Step 2:

Define a cut-off function to localize the integration region.

By a standard approximation argument, the validity of (37) extends to all Lipschitz functions $\eta$ having compact support. In order to further analyze the integral equation, for $0<ϵ\le 1$, we define

$$\begin{array}{c}\hfill {\eta }_{ϵ}\left(t\right)=\left\{\begin{array}{cc}1,\hfill & t\le 1,\hfill \\ 1+{\displaystyle \frac{1-t}{ϵ}},\hfill & 1<t<1+ϵ,\hfill \\ 0,\hfill & t\ge 1+ϵ,\hfill \end{array}\right.\end{array}$$

(38)

and take $\eta \left(r\right(P\left)\right)$ as follows:

$$\begin{array}{c}\hfill \eta \left(r\left(P\right)\right)={\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right),R>2R_1.\end{array}$$

(39)

From (38) and (39), and letting $R_2=2R_1$, we have

$$\begin{array}{c}\hfill \eta \left(r\left(P\right)\right)={\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right)=\left\{\begin{array}{cc}0,\hfill & r\left(P\right)\le R_1,\hfill \\ 1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right),\hfill & R_1<r\left(P\right)\le R_2,\hfill \\ 1,\hfill & R_2<r\left(P\right)\le R,\hfill \\ {\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right),\hfill & R<r\left(P\right)<(1+ϵ)R,\hfill \\ 0,\hfill & r\left(P\right)\ge (1+ϵ)R.\hfill \end{array}\right.\end{array}$$

(40)

Substituting (40) into (37), we can obtain each term on the left-hand side of Equation (37) as follows:

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}\eta \left(r\left(P\right)\right)d{v}_g\hfill \\ & =& {\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right)d{v}_g\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}d{v}_g\hfill \\ & & +{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}{\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)d{v}_g\hfill \end{array}$$

(41)

and

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}\eta \left(r\left(P\right)\right)d{v}_g\hfill \\ & =& {\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right)d{v}_g\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}d{v}_g\hfill \\ & & +{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}{\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)d{v}_g\hfill \end{array}$$

(42)

and

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial \eta \left(r\right(P\left)\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & =& {\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial \left(1-{\eta }_1\left(\frac{r\left(P\right)}{R_1}\right)\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & & +{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial {\eta }_{ϵ}\left(\frac{r\left(P\right)}{R}\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & =& -{\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial {\eta }_1\left(\frac{r\left(P\right)}{R_1}\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & & -{\displaystyle \frac{1}{Rϵ}}{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }d{v}_g,\hfill \end{array}$$

(43)

similarly,

$$\begin{array}{ccc}\hfill & & {\int }_{M\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial \eta \left(r\right(P\left)\right)}{\partial x_l}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\tilde{F}}_{\beta }d{v}_g\hfill \\ & =& -{\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial {\eta }_1\left(\frac{r\left(P\right)}{R_1}\right)}{\partial x_l}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\tilde{F}}_{\beta }d{v}_g\hfill \\ & & -{\displaystyle \frac{1}{Rϵ}}{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }d{v}_g.\hfill \end{array}$$

(44)

Letting $ϵ\to 0$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{Rϵ}}{\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }d{v}_g\\ \hfill \to {\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }d{s}_g\end{array}$$

(45)

and

$$\begin{array}{c}\hfill {\int }_{B\left(\right(1+ϵ\left)R\right)\setminus B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}{\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)d{v}_g\\ \hfill \to ϵR{\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}{\eta }_{ϵ}\left({\displaystyle \frac{r\left(P\right)}{R}}\right)d{s}_g=0.\end{array}$$

(46)

From (37)–(46), we have

$$\begin{array}{ccc}\hfill & & {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}d{v}_g\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}d{v}_g+\Phi \left(R_1\right)\hfill \\ & =& {\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }d{s}_g\hfill \\ & & +{\int }_{\partial B\left(R\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }d{s}_g,\hfill \end{array}$$

(47)

where

$$\begin{array}{ccc}\hfill \Phi \left(R_1\right)& =& {\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right)d{v}_g\hfill \\ & & -{\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial {\eta }_1\left(\frac{r\left(P\right)}{R_1}\right)}{\partial x_j}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }d{v}_g\hfill \\ & & +{\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial {\tilde{F}}_{\beta }}{\partial x_l}}\left(1-{\eta }_1\left({\displaystyle \frac{r\left(P\right)}{R_1}}\right)\right)d{v}_g\hfill \\ & & -{\int }_{B\left(R_2\right)\setminus B\left(R_1\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial {\eta }_1\left(\frac{r\left(P\right)}{R_1}\right)}{\partial x_l}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\tilde{F}}_{\beta }d{v}_g\hfill \\ & :=& {\Phi }_p\left(R_1\right)+{\Phi }_q\left(R_1\right).\hfill \end{array}$$

(48)

Note that ${\tilde{F}}_{\alpha }={\displaystyle \frac{F_{\alpha }^2-{\epsilon }_{\alpha }^2}{F_{\alpha }}}$. Thus, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\tilde{F}}_{\alpha }}{\partial x_j}}=\left(1+{\displaystyle \frac{{\epsilon }_{\alpha }^2}{F_{\alpha }^2}}\right){\displaystyle \frac{\partial F_{\alpha }}{\partial x_j}}.\end{array}$$

(49)

From (47) and (49), we have

$$\begin{array}{ccc}\hfill & & {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_j}}\left(1+{\displaystyle \frac{{\epsilon }_{\alpha }^2}{F_{\alpha }^2}}\right)d{v}_g\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_l}}\left(1+{\displaystyle \frac{{\epsilon }_{\beta }^2}{F_{\beta }^2}}\right)d{v}_g+\Phi \left(R_1\right)\hfill \\ & =& {\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }d{s}_g\hfill \\ & & +{\int }_{\partial B\left(R\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }d{s}_g.\hfill \end{array}$$

(50)

Since the term

$${\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }+{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }$$

is coordinate-independent. At any boundary point $P\in \partial B\left(R\right)$, we simplify the calculations using the adapted coordinates, where $g_{ij}\left(P\right)={\delta }_{ij}$, $g^{ij}\left(P\right)={\delta }^{ij}$. Using ${\left|Dr\right|}^2=1$, we evaluate the expression at P.

$$\begin{array}{ccc}\hfill & & {\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }+{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j=1}^m\sum _{\alpha =1}^n{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,\alpha }{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_i}}{\tilde{F}}_{\alpha }\hfill \\ & \le & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^m{\left(\sum _{\alpha =1}^n{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }{\displaystyle \frac{\partial r\left(P\right)}{\partial x_i}}\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^m{\left(\sum _{\alpha =1}^n{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}{\left[\sum _{i=1}^m{\left({\displaystyle \frac{\partial r\left(P\right)}{\partial x_i}}\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^m{\left(\sum _{\alpha =1}^n{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\tilde{F}}_{\alpha }\right)}^2\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^m\left(\sum _{\alpha =1}^n{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}\right)\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^m\left(\sum _{\alpha =1}^n{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}\right)\right]}^{{\displaystyle \frac{1}{2}}}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right){\left[\sum _{i=1}^{m}h\left(dF\left({\displaystyle \frac{\partial }{\partial x_i}}\right),dF\left({\displaystyle \frac{\partial }{\partial x_i}}\right)\right)\right]}^{{\displaystyle \frac{1}{2}}}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& \left({\left|dF\right|}^{p-1}+{\left|dF\right|}^{q-1}\right){\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(51)

thus

$$\begin{array}{ccc}\hfill & & {\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_j}}{\tilde{F}}_{\alpha }+{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial r\left(P\right)}{\partial x_l}}{\tilde{F}}_{\beta }d{s}_g\hfill \\ & \le & {\int }_{\partial B\left(R\right)}\left({\left|dF\right|}^{p-1}+{\left|dF\right|}^{q-1}\right){\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}d{s}_g\hfill \\ & \le & {\left[{\int }_{\partial B\left(R\right)}{\left|dF\right|}^{p}d{s}_g\right]}^{{\displaystyle \frac{p-1}{p}}}{\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{p}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{p}}}\hfill \\ & & +{\left[{\int }_{\partial B\left(R\right)}{\left|dF\right|}^{q}d{s}_g\right]}^{{\displaystyle \frac{q-1}{q}}}{\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{q}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

(52)

On the other hand, we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_j}}\left(1+{\displaystyle \frac{{\epsilon }_{\alpha }^2}{F_{\alpha }^2}}\right)d{v}_g\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_l}}\left(1+{\displaystyle \frac{{\epsilon }_{\beta }^2}{F_{\beta }^2}}\right)d{v}_g+\Phi \left(R_1\right)\hfill \\ & \ge & {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p-2}\sum _{i,j,\alpha }{g}^{ij}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_i}}{\displaystyle \frac{\partial F_{\alpha }}{\partial x_j}}d{v}_g+{\Phi }_p\left(R_1\right)\hfill \\ & & +{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q-2}\sum _{k,l,\beta }{g}^{kl}{\displaystyle \frac{\partial F_{\beta }}{\partial x_k}}{\displaystyle \frac{\partial F_{\beta }}{\partial x_l}}d{v}_g+{\Phi }_q\left(R_1\right)\hfill \\ & =& {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p}d{v}_g+{\Phi }_p\left(R_1\right)+{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q}d{v}_g+{\Phi }_q\left(R_1\right).\hfill \end{array}$$

(53)

Step 3:

Construct an auxiliary function to facilitate calculations.

Setting

$$\begin{array}{ccc}\hfill \Psi \left(R\right)& =& {\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{p}d{v}_g+{\Phi }_p\left(R_1\right)+{\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{q}d{v}_g+{\Phi }_q\left(R_1\right)\hfill \\ & :=& {\Psi }_p\left(R\right)+{\Psi }_q\left(R\right),\hfill \end{array}$$

(54)

then, we have

$${\Psi }_y\left(R\right)-{\Phi }_y\left(R_1\right)={\int }_{B\left(R\right)\setminus B\left(R_2\right)}{\left|dF\right|}^{y}d{v}_g,$$

and

$${\Psi }_y^{\prime }\left(R\right)={\displaystyle \frac{d}{dR}}{\Psi }_y\left(R\right)={\int }_{\partial B\left(R\right)}{\left|dF\right|}^{y}d{s}_g.$$

(55)

Therefore,

$${\Psi }^{\prime }\left(R\right)={\displaystyle \frac{d}{dR}}\Psi \left(R\right)={\int }_{\partial B\left(R\right)}\left({\left|dF\right|}^p+{\left|dF\right|}^q\right)d{s}_g.$$

(56)

Combining (50), (52)–(55), we have

$$\begin{array}{ccc}\hfill \Psi \left(R\right)& =& {\Psi }_p\left(R\right)+{\Psi }_q\left(R\right)\hfill \\ & \le & {\left[{\Psi }_p^{\prime }\left(R\right)\right]}^{{\displaystyle \frac{p-1}{p}}}{\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{p}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{p}}}+{\left[{\Psi }_q^{\prime }\left(R\right)\right]}^{{\displaystyle \frac{q-1}{q}}}{\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{q}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Given that $E_{p,q}^R\left(F\right)=\infty$, it follows from the definition of $\Psi \left(R\right)$ that there exists $R_3\ge R_2$ such that ${\Psi }_y\left(R\right)>0$ for all $R>R_3$. Set

$$M\left(R\right)={\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{p}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{p-1}}}+{\left[{\int }_{\partial B\left(R\right)}{\left(\sum _{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{{\displaystyle \frac{q}{2}}}d{s}_g\right]}^{{\displaystyle \frac{1}{q-1}}}:=M_p\left(R\right)+M_q\left(R\right).$$

Then

$$\begin{array}{c}\hfill {\left[{\Psi }_y\left(R\right)\right]}^{{\displaystyle \frac{y}{y-1}}}\le {\Psi }_y^{\prime }\left(R\right)M_y\left(R\right).\end{array}$$

(57)

Step 4:

Restore the original global properties through a limiting process.

For any $R_4>R>R_3$, the following holds:

$${\int }_R^{R_4}{\displaystyle \frac{{\Psi }_y^{\prime }\left(r\right)}{{\left[{\Psi }_y\left(r\right)\right]}^{\frac{y}{y-1}}}}dr\ge {\int }_R^{R_4}{\displaystyle \frac{1}{M_y\left(r\right)}}dr.$$

Letting $R_4\to \infty$ and using ${\Psi }_y\left(R\right)>0$,

$${\displaystyle \frac{1}{{\left[{\Psi }_y\left(R\right)\right]}^{\frac{1}{y-1}}}}\ge {\displaystyle \frac{1}{y-1}}{\int }_R^{\infty }{\displaystyle \frac{1}{M_y\left(r\right)}}dr,$$

so

$$\begin{array}{c}\hfill \Psi \left(R\right)={\Psi }_p\left(R\right)+{\Psi }_q\left(R\right)\le {(p-1)}^{p-1}{\left[{\displaystyle \frac{1}{{\int }_R^{\infty }\frac{1}{M\left(r\right)}dr}}\right]}^{p-1}+{(q-1)}^{q-1}{\left[{\displaystyle \frac{1}{{\int }_R^{\infty }\frac{1}{M\left(r\right)}dr}}\right]}^{q-1},\end{array}$$

(58)

where $R>R_3$. Since $F\left(P\right)\to X_0$ as $r\left(P\right)\to \infty$,

$$\begin{array}{c}\hfill M_y\left(R\right)\le {\left[{\int }_{\partial B\left(R\right)}{\varphi }^y\left(R\right)d{s}_g\right]}^{{\displaystyle \frac{1}{y-1}}}={\left[\varphi \left(R\right)\right]}^{{\displaystyle \frac{y}{y-1}}}{\left[vol(\partial B\left(R\right))\right]}^{{\displaystyle \frac{1}{y-1}}},\end{array}$$

(59)

where $\varphi \left(R\right)$ is a function satisfying the following conditions:

(1)

$\varphi \left(R\right)$ is non-increasing on $(R_3,\infty )$ and ${lim}_{R\to \infty }\varphi \left(R\right)=0$;

(2)

$\varphi \left(R\right)\ge {max}_{r\left(P\right)=R}\left[{\left({\sum }_{\alpha =1}^n{\left({\tilde{F}}_{\alpha }\right)}^2\right)}^{1/2}\right]$;

(3)

${\left[\varphi \left(R\right)\right]}^{{\displaystyle \frac{y}{y-1}}}\le C\left(y\right)R^{{\displaystyle \frac{\tilde{\varrho }}{y-1}}}{\int }_R^{\infty }{\displaystyle \frac{dr}{{\left[vol(\partial B\left(r\right))\right]}^{\frac{1}{y-1}}}}$.

Here, $C\left(y\right)$ is a constant determined by $X_0$, so we get

$${\int }_R^{\infty }{\displaystyle \frac{1}{M_y\left(r\right)}}dr\ge {\displaystyle \frac{1}{{\left[\varphi \left(R\right)\right]}^{\frac{y}{y-1}}}}{\int }_R^{\infty }{\displaystyle \frac{1}{{\left[vol(\partial B\left(r\right))\right]}^{\frac{1}{y-1}}}}dr\ge {\displaystyle \frac{1}{C\left(y\right)R^{\frac{\tilde{\varrho }}{y-1}}}}.$$

(60)

Then, we have

$$\begin{array}{c}\hfill {\Psi }_y\left(R\right)\le {\left(C\left(y\right)(y-1)\right)}^{y-1}{R}^{\tilde{\varrho }},\mathrm{for}R>R_3.\end{array}$$

(61)

In conclusion, we obtain

$$\begin{array}{ccc}\hfill E_{p,q}^R\left(F\right)& =& {\int }_{B\left(R\right)\setminus B\left(R_2\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g+{\int }_{B\left(R_2\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g\hfill \\ & =& {\displaystyle \frac{{\Psi }_p\left(R\right)}{p}}+{\displaystyle \frac{{\Psi }_q\left(R\right)}{q}}+{\int }_{B\left(R_2\right)}\left({\displaystyle \frac{{\left|dF\right|}^p}{p}}+{\displaystyle \frac{{\left|dF\right|}^q}{q}}\right)d{v}_g-{\displaystyle \frac{{\Phi }_p\left(R_1\right)}{p}}-{\displaystyle \frac{{\Phi }_q\left(R_1\right)}{q}}\hfill \\ & \le & \left(C(p,q)R^{\tilde{\varrho }-\varrho }+{\displaystyle \frac{c\left(F\right)}{R^{\varrho }}}\right)R^{\varrho },\hfill \end{array}$$

where $C(p,q)={\left(C\left(p\right)(p-1)\right)}^{p-1}/p+{\left(C\left(q\right)(q-1)\right)}^{q-1}/q$. □

Remark 4. 

(1) Let M be a complete noncompact Riemannian manifold with non-negative Ricci curvature and N be a Riemannian manifold with non-positive sectional curvature. Cheng [5] used gradient estimates to show that any harmonic map $F:M\to N$ with compact image is constant. Takeuchi [21] further proved that each p-harmonic map F satisfying ${\int }_M{\left|dF\right|}^{2p-2}d{v}_g<\infty$ must be constant. (2) Let M be a complete noncompact Riemannian m-manifold ($m\ge 2$) supporting a weighted Poincaré inequality with Ricci curvature $Ri{c}_M\ge -\tau \rho \left(x\right)$, where $\rho \left(x\right)$ is a positive function. Chang-Chen-Wei [4] proved that every weakly p-harmonic function with finite p-energy $E_p$ is constant, and a strongly p-harmonic function with finite q-energy $E_q$ is constant.

Corollary 3. 

Consider a $(p,q)$-harmonic map $F:(M^m,g)\to ({\mathbb{R}}^n,h)$, where $m>\{p,q\}$. For the curvature tensor $K_r$ of M, assume the following conditions:

(1) ${\displaystyle \frac{B}{{(1+r^2)}^{1+ϵ}}}\ge K_r\ge -{\displaystyle \frac{A}{{(1+r^2)}^{1+ϵ}}}$ for some constants $ϵ>0,A\ge 0,B\in [0,2ϵ]$ with $1+(m-1)\left(1-{\displaystyle \frac{B}{2ϵ}}\right)-max\{p,q\}{e}^{{\displaystyle \frac{A}{2ϵ}}}>\tilde{\varrho }>0$, and

$${\left[\underset{r\left(P\right)=R}{max}{h}^2(F\left(P\right),X_0)\right]}^2\le {\left[{\displaystyle \frac{y-1}{m-y}}\right]}^{y-1}{\displaystyle \frac{R^{\tilde{\varrho }-(m-y)}}{v_m{e}^{\frac{(m-1)A}{2ϵ}}}},$$

where $v_m$ is the $(m-1)$-volume of the unit sphere in ${\mathbb{R}}^m$.

Or (2) ${\displaystyle \frac{b^2}{1+r^2}}\ge K_r\ge -{\displaystyle \frac{a^2}{1+r^2}}$ for some constants $a\ge 0,b^2\in [0,{\displaystyle \frac{1}{4}}]$ with $1+(m-1){\displaystyle \frac{1+\sqrt{1-4b^2}}{2}}-max\{p,q\}{\displaystyle \frac{1+\sqrt{1+4a^2}}{2}}>\tilde{\varrho }>0,$ and

$${\left[\underset{r\left(P\right)=R}{max}{h}^2(F\left(P\right),X_0)\right]}^2\le C^{-1}{\left[{\displaystyle \frac{y-1}{(m-1)\frac{1+\sqrt{1+4a^2}-(y-1)}{2}}}\right]}^{y-1}{R}^{\tilde{\varrho }-(m-1){\displaystyle \frac{1+\sqrt{1+4a^2}}{2}}+y-1},$$

If $r\left(P\right)\to \infty$, $F\left(P\right)\to X_0\in S^{n-1}$, then F is necessarily constant.

Proof. 

For case (1), by the assumption, the Ricci curvature satisfies the lower bound

$${\mathrm{Ric}}^M\left(P\right)\ge -{\displaystyle \frac{(m-1)A}{{(1+r^2\left(P\right))}^{1+ϵ}}},\forall P\in M^m.$$

Since

$${\int }_0^{\infty }{\displaystyle \frac{Ar}{{(1+r^2)}^{1+ϵ}}}dr={\displaystyle \frac{A}{2ϵ}},$$

the volume comparison theorem (cf. [28]) yields

$$\mathrm{vol}(\partial B\left(R\right))\le v_m{e}^{{\displaystyle \frac{(m-1)A}{2ϵ}}}{R}^{m-1}.$$

By combining the previous volume estimate and some integral inequalities, we can show that

$${\left[{\int }_R^{\infty }{\displaystyle \frac{dr}{{\left(\mathrm{vol}(\partial B\left(r\right))\right)}^{\frac{1}{y-1}}}}\right]}^{-(y-1)}\le {\left[{\displaystyle \frac{m-y}{y-1}}\right]}^{y-1}{v}_m{e}^{{\displaystyle \frac{(m-1)A}{2ϵ}}}{R}^{m-y},R\gg 1.$$

Similarly, for case (2), we have

$$\mathrm{vol}(\partial B\left(R\right))\le C{R}^{(m-1){\displaystyle \frac{1+\sqrt{1+4a^2}}{2}}},$$

and

$${\mathrm{Ric}}^M\left(P\right)\ge -{\displaystyle \frac{(m-1)a^2}{1+r^2\left(P\right)}},\forall P\in M^m.$$

Therefore, for $R\gg 1$, we have

$${\left[{\int }_R^{\infty }{\displaystyle \frac{dr}{{\left(\mathrm{vol}(\partial B\left(r\right))\right)}^{\frac{1}{y-1}}}}\right]}^{-(y-1)}\le C{\left[{\displaystyle \frac{(m-1)\frac{1+\sqrt{1+4a^2}}{2}-(y-1)}{y-1}}\right]}^{y-1}{R}^{(m-1){\displaystyle \frac{1+\sqrt{1+4a^2}}{2}}-(y-1)},$$

where C is a constant. The conclusion follows by applying Corollary 2 and Theorem 6. □

5. Examples of (p,q)-SSU Manifolds

In this section, we obtain some examples of $(p,q)$-$SSU$ manifolds.

Theorem 7. 

Let $M^m\subset {\mathbb{R}}^{m+1}$, $(m\ge max\{p,q\left\}\right)$ be the compact hypersurface. Assume that the principal curvatures ${\mu }_i$ of $M^m$ satisfy $0<{\mu }_1\le {\mu }_2\le \cdots \le {\mu }_m$ and ${\mu }_m<{\displaystyle \frac{1}{max\left\{\right(p-1),(q-1\left)\right\}}}[{\mu }_1+{\mu }_2+\cdots +{\mu }_{m-1}]$. Then M is $(p,q)$-$SSU$ manifold.

Proof. 

Similar to the proof of Theorem 3.3 in [29], using the definition of the $(p,q)$-$SSU$ manifolds and taking $z=e_j$, we have

$$\begin{array}{ccc}& & max\left\{\right(p-2),(q-2\left)\right\}\langle \mathcal{H}(z,z),\mathcal{H}(z,z)\rangle \hfill \\ & & +\sum _{i=1}^m\left(2〈\mathcal{H}(z,e_i),\mathcal{H}(z,e_i)〉-〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\right)\hfill \\ & =& max\{(p-2),(q-2)\}{\mu }_j^2+\left(2{\mu }_j^2-{\mu }_j\sum _{i=1}^m{\mu }_i\right)\hfill \\ & \le & {\mu }_j\left(max\{(p-1),(q-1)\}{\mu }_m-\sum _{i=1}^{m-1}{\mu }_i\right)<0.\hfill \end{array}$$

This completes the proof. □

Using Theorem 7, we have the following corollary.

Corollary 4. 

The standard sphere $S^m$ is $(p,q)$-$SSU$ if and only if $m>max\{p,q\}$.

Proof. 

$S^m$ is a compact convex hypersurface in ${\mathbb{R}}^{m+1}$. By Theorem 7 and its principal curvatures satisfy

$${\mu }_1={\mu }_2=\cdots ={\mu }_m=1.$$

That is, $m>max\{p,q\}$. This completes the proof. □

Corollary 5. 

The graph of $f\left(x\right)=x_1^2+\cdots +x_m^2$, $x=(x_1,\cdots ,x_m)\in {\mathbb{R}}^m$ is $(p,q)$-$SSU$ if and only if $m>max\{p,q\}$.

Theorem 8. 

Let $\tilde{M}\in {\mathbb{R}}^d$ be a compact convex hypersurface and its principal curvatures satisfy

$$0<{\mu }_1\le {\mu }_2\le \cdots \le {\mu }_{d-1}.$$

If

$$〈Ri{c}^M\left(z\right),z〉>{\displaystyle \frac{max\left\{\right(p-1),(q-1\left)\right\}}{max\{p,q\}}}k{\mu }_{d-1}^2,$$

then M is $(p,q)$-$SSU$ manifold, where $M\in \tilde{M}$ is a compact connected minimal k-submanifold and z is any unit tangent vector to M.

Proof. 

Let $\mathcal{H}$, ${\mathcal{H}}_1$, and $\tilde{\mathcal{H}}$ denote the second fundamental form of $M\in {\mathbb{R}}^d$, $M\in \tilde{M}$, and $\tilde{M}\in {\mathbb{R}}^d$, respectively. According to the Gauss equation, we obtain

$$\mathcal{H}(Y,Z)={\mathcal{H}}_1(Y,Z)+\tilde{\mathcal{H}}(Y,Z)\nu ,$$

where ν is the unit normal field of $\tilde{M}\in {\mathbb{R}}^d$. By the definition of minimal k-submanifold, we have

$$\sum _{i=1}^k\mathcal{H}(e_i,e_i)=\sum _{i=1}^k{\mathcal{H}}_1(e_i,e_i)+\sum _{i=1}^k\tilde{\mathcal{H}}(e_i,e_i)\nu =\sum _{i=1}^k\tilde{\mathcal{H}}(e_i,e_i)\nu ,$$

where ${\left\{e_i\right\}}_{i=1}^k$ is a local orthonormal frame on M. Let $\tilde{\mathcal{H}}(e_i,e_j)={\mu }_i{\delta }_{ij}.$ Hence,

$$\begin{array}{ccc}& & max\left\{\right(p-2),(q-2\left)\right\}\langle \mathcal{H}(z,z),\mathcal{H}(z,z)\rangle \hfill \\ & & +\sum _{i=1}^k\left(2〈\mathcal{H}(z,e_i),\mathcal{H}(z,e_i)〉-〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\right)\hfill \\ & \le & max\{(p-2),(q-2)\}\sum _{i=1}^k\langle \mathcal{H}(z,e_i),\mathcal{H}(z,e_i)\rangle \hfill \\ & & +\sum _{i=1}^k\left(2〈\mathcal{H}(z,e_i),\mathcal{H}(z,e_i)〉-〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\right)\hfill \\ & =& \sum _{i=1}^k\left(max\{p,q\}〈\mathcal{H}(z,e_i),\mathcal{H}(z,e_i)〉-〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\right)\hfill \\ & =& -max\{p,q\}〈Ri{c}^M\left(z\right),z〉+\sum _{i=1}^{k}max\{p,q\}〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉-\sum _{i=1}^k〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\hfill \\ & =& -max\{p,q\}〈Ri{c}^M\left(z\right),z〉+\sum _{i=1}^{k}max\{(p-1),(q-1)\}〈\mathcal{H}(z,z),\mathcal{H}(e_i,e_i)〉\hfill \\ & =& -max\{p,q\}〈Ri{c}^M\left(z\right),z〉+max\{(p-1),(q-1)\}\sum _{i=1}^k〈\tilde{\mathcal{H}}(z,z),\tilde{\mathcal{H}}(e_i,e_i)〉\hfill \\ & \le & -max\{p,q\}〈Ri{c}^M\left(z\right),z〉+max\{(p-1),(q-1)\}\sum _{i=1}^k{\mu }_i{\mu }_{d-1}\hfill \\ & \le & -max\{p,q\}〈Ri{c}^M\left(z\right),z〉+max\{(p-1),(q-1)\}k{\mu }_{d-1}^2\hfill \\ & <& 0,\hfill \end{array}$$

(62)

where we use the Gauss equation. This completes the proof. □

6. Stability of (p,q)-Harmonic Maps

We begin by recalling key concepts in submanifold geometry (cf. [12]), which will underpin the subsequent results.

Consider an isometric immersion $M^m\hookrightarrow {\mathbb{R}}^d$. Let D and $\overline{D}$ denote the Riemannian connection on M and the standard flat connection on ${\mathbb{R}}^d$, respectively. The second fundamental form $\mathcal{H}$ relates these connections via

$${\overline{D}}_{Y}Z=D_{Y}Z+\mathcal{H}(Y,Z),\mathrm{for}Y,Z\in \Gamma \left(TM\right).$$

Let $T^{\perp }M$ denote the normal bundle of M in ${\mathbb{R}}^d$. Given a smooth section $X\in T^{\perp }M$, the Weingarten map ${\mathcal{A}}^{\xi }$ and $\mathcal{H}$ satisfy

$$\begin{array}{c}\hfill \langle {\mathcal{A}}^{\xi }Y,Z\rangle =\langle \mathcal{H}(Y,Z),\xi \rangle .\end{array}$$

(63)

Fix a point $P\in M$, and let ${\left\{e_{\alpha }\right\}}_{\alpha =m+1}^d$ be an orthonormal basis for $T_P^{\perp }M$. The Ricci tensor ${Ric}^M:T_P\left(M\right)\to T_P\left(M\right)$ is given by

$$Ri{c}^M\left(z\right)=\sum _{i=1}^{m}R(z,e_i)e_i,z\in T_P\left(M\right).$$

The Gauss equation then yields

$$\begin{array}{c}\hfill Ri{c}^M=\sum _{\alpha =m+1}^d\mathrm{trace}\left({\mathcal{A}}^{\alpha }\right){\mathcal{A}}^{\alpha }-\sum _{\alpha =m+1}^d{\mathcal{A}}^{\alpha }{\mathcal{A}}^{\alpha }.\end{array}$$

(64)

For the subsequent proof, we require the following Weitzenböck formula:

Lemma 1 

([30]). Let ω be an r-form with values in a vector bundle. Then

$$\Delta \omega =-{\nabla }^2\omega +S,$$

where ${\nabla }^2$ denotes the trace-Laplace operator introduced in the last section and for any $Y_1,\cdots ,Y_r\in \Gamma \left(TM\right)$

$$\begin{array}{c}\hfill S(Y_1,\cdots ,Y_r)=\sum _{a=1}^r{(-1)}^a\left(R(e_i,Y_a)\omega \right)(e_i,Y_1,\cdots ,{\widehat{Y}}_a,\cdots ,Y_r).\end{array}$$

Theorem 9. 

Suppose that $F:M\to N$ is a $(p,q)$-harmonic map, where M is a compact Riemannian manifold and N is a compact $(p,q)$-$SSU$ manifold. Then, a non-constant stable map F does not exist.

Proof. 

Consider a local orthonormal frame field ${\left\{e_i\right\}}_{i=1}^d$ for ${\mathbb{R}}^d$, such that $\{e_1,\cdots ,e_n\}$ are tangent to $N^n$, and ${\left\{e_{\alpha }\right\}}_{\alpha =n+1}^d$ are normal. At a fixed point $P_0\in N^n$, we impose the local condition $D_{e_i}{e}_j{|}_{P_0}=0$. Fix an orthonormal basis ${\left\{E_A\right\}}_{A=1}^d$ for ${\mathbb{R}}^d$, and define

$$\begin{array}{c}\hfill W_A=\sum _{i=1}^n{W}_A^i{e}_i,W_A^i=\langle E_A,e_i\rangle ,W_A^{\alpha }=\langle E_A,e_{\alpha }\rangle .\end{array}$$

(65)

The index ranges are as follows: $i,j,k,l,s,t\in \{1,\cdots ,n\},a,b\in \{1,\cdots ,m\},\alpha ,\beta ,\cdots \in \{n+1,\cdots ,d\},A\in \{1,\cdots ,d\}$. Taking $V=W=W_A$ in (13), we have

$$\begin{array}{ccc}\hfill & & \sum _{A}I(W_A,W_A)\hfill \\ & =& (p-2){\int }_M{\left|dF\right|}^{p-4}\sum _{a,b,A}\langle {\tilde{D}}_{{ϵ}_a}{W}_A,dF\left({ϵ}_a\right)\rangle \langle {\tilde{D}}_{{ϵ}_b}{W}_A,dF\left({ϵ}_b\right)\rangle d{v}_g\hfill \\ & & +(q-2){\int }_M{\left|dF\right|}^{q-4}\sum _{a,b,A}\langle {\tilde{D}}_{{ϵ}_a}{W}_A,dF\left({ϵ}_a\right)\rangle \langle {\tilde{D}}_{{ϵ}_b}{W}_A,dF\left({ϵ}_b\right)\rangle d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A}h({\tilde{D}}_{{ϵ}_a}{W}_A,{\tilde{D}}_{{ϵ}_a}{W}_A)d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A}h(R^N(W_A,dF\left({ϵ}_a\right))W_A,dF\left({ϵ}_a\right))d{v}_g,\hfill \end{array}$$

(66)

where ${\left\{{ϵ}_a\right\}}_{a=1}^m$ is a locally orthogonal frame field on $M^m$. Using the components ${\mathcal{H}}_{ij}^{\alpha }=\langle \mathcal{H}(e_i,e_j),e_{\alpha }\rangle$ of the second fundamental form of $N^n$ in $R^d$ and defining $dF\left({ϵ}_a\right)={\sum }_i{F}_a^i{e}_i$, we observe that the product matrix ${\left(F_a^i\right)}^{\top }·\left(F_a^i\right)$ is symmetric:

$$\begin{array}{c}\hfill {\left(\sum _{a=1}^m{F}_a^i{F}_a^j\right)}_{i,j=1,\cdots ,n},\end{array}$$

where ${\left(F_a^i\right)}^{\top }$ is the transpose of $\left(F_a^i\right)$. Consequently, at $P_0$, we choose $\{e_1,\cdots ,e_n\}$ to diagonalize this matrix, so that

$$\begin{array}{c}\hfill \sum _{a=1}^m{F}_a^i{F}_a^j={\mu }_i^2{\delta }_{ij}.\end{array}$$

(67)

The covariant derivatives simplify to

$$\begin{array}{c}\hfill {\tilde{D}}_{{ϵ}_a}{W}_A{{=}^{N^n}D}_{dF\left({ϵ}_a\right)}{W}_A={\sum _i}^{N^n}{D}_{e_i}{W}_A·F_a^i,D_{e_i}{W}_A=W_A^{\alpha }{\mathcal{H}}_{ik}^{\alpha }{e}_k.\end{array}$$

(68)

From Equation (68), we see that each term on the right-hand side of Equation (66) is, respectively, given by

$$\begin{array}{ccc}\hfill & & {\left|dF\right|}^{p-4}\sum _{a,b,A}\langle {\tilde{D}}_{{ϵ}_a}{W}_A,dF\left({ϵ}_a\right)\rangle \langle {\tilde{D}}_{{ϵ}_b}{W}_A,dF\left({ϵ}_b\right)\rangle \hfill \\ & =& {\left|dF\right|}^{p-4}\sum _{a,b,A,i,j,s,t}{F}_a^{i}h(D_{e_i}{W}_A,F_a^s{e}_s)F_b^{j}h(D_{e_j}{W}_A,F_b^t{e}_t)\hfill \\ & =& {\left|dF\right|}^{p-4}\sum _{a,b,A,i,j,k,l,s,t,\alpha ,\beta }{F}_a^i{W}_A^{\alpha }{\mathcal{H}}_{ik}^{\alpha }{F}_a^{s}h(e_k,e_s)F_b^j{W}_A^{\beta }{\mathcal{H}}_{jl}^{\beta }{F}_b^{t}h(e_l,e_t)\hfill \\ & =& {\left|dF\right|}^{p-4}\sum _{i,j,k,l,s,t}\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_j,e_l)\rangle {\mu }_i^2{\delta }_{is}{\mu }_j^2{\delta }_{jt}{\delta }_{ks}{\delta }_{lt}\hfill \\ & =& {\left|dF\right|}^{p-4}\sum _{i,j}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle {\mu }_i^2{\mu }_j^2\hfill \\ & \le & {\left|dF\right|}^{p-2}\sum _i\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle {\mu }_i^2\hfill \end{array}$$

(69)

and

$$\begin{array}{ccc}\hfill & & {\left|dF\right|}^{q-4}\sum _{a,b,A}h({\tilde{D}}_{{ϵ}_a}{W}_A,dF\left({ϵ}_a\right))h({\tilde{D}}_{{ϵ}_b}{W}_A,dF\left({ϵ}_b\right))\hfill \\ & \le & {\left|dF\right|}^{q-2}\sum _i\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle {\mu }_i^2\hfill \end{array}$$

(70)

and

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A}h({\tilde{D}}_{{ϵ}_a}{W}_A,{\tilde{D}}_{{ϵ}_a}{W}_A)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,i,j,A}{F}_a^i{F}_a^{j}h(D_{e_i}{W}_A,D_{e_j}{W}_A)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{\alpha ,\beta ,A,i,j,k,l,a}{F}_a^i{F}_a^{j}h(W_A^{\alpha }{\mathcal{H}}_{ik}^{\alpha }{e}_k,W_A^{\beta }{\mathcal{H}}_{jl}^{\beta }{e}_l)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j,k,l}{\mu }_i^2{\delta }_{ij}{\delta }_{kl}\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_j,e_l)\rangle \hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j}{\mu }_i^2\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle \hfill \end{array}$$

(71)

and

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A}^{m}h(R^N(W_A,dF\left({ϵ}_a\right))W_A,dF\left({ϵ}_a\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A,i,j,k,l}{W}_A^i{W}_A^j{F}_a^k{F}_a^{l}h(R^N(e_i,e_k)e_j,e_l)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{a,A,i,j,k,l}{W}_A^i{W}_A^j{F}_a^k{F}_a^l[\langle \mathcal{H}(e_j,e_k),\mathcal{H}(e_i,e_l)\rangle -\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_k,e_l)\rangle ]\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k}[\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_i,e_k)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_k,e_k)\rangle ]{\mu }_k^2\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j}[\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle ]{\mu }_i^2.\hfill \end{array}$$

(72)

By (66)–(72), we can obtain

$$\begin{array}{ccc}\hfill \sum _{A}I(W_A,W_A)& \le & {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _i{\mu }_i^2\{max\{(p-2),(q-2)\}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle \hfill \\ & & +\sum _j(2\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle )\}d{v}_g.\hfill \end{array}$$

Since N is a $(p,q)$-$SSU$ manifold, we see that if F is not constant, then

$$\sum _{A}I(W_A,W_A)<0.$$

This implies that F cannot be a stable $(p,q)$-harmonic map. Hence, a stable $(p,q)$-harmonic map must be constant. □

Remark 5. 

Take the target manifold as a complete orientable hypersurface in ${\mathbb{R}}^{n+1}$. When $q=p$, condition

$$\begin{array}{ccc}& & max\{(p-2),(q-2)\}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle \hfill \\ & & +\sum _j(2\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle )\hfill \end{array}$$

becomes

$$\begin{array}{c}\hfill (p-1){\lambda }^2<(n-1)K\left(y\right),\end{array}$$

(73)

where $K\left(y\right)$ is the minimum of all sectional curvatures at $y\in N$, ${\lambda }_i$ is the principal curvatures of N, and ${\lambda }^2=max{\left\{{\lambda }_i^2\right\}}_{i=1}^n$. In [21], Takeuchi proved that under condition (73) above, each stable p-harmonic map from any compact Riemannian manifold into the hypersurface is constant.

Theorem 10. 

Suppose that $F:M\to N$ is a $(p,q)$-harmonic map, where M is a compact $(p,q)$-$SSU$ manifold and N is a Riemannian manifold. Then a non-constant stable map F does not exist.

Proof. 

We take a local orthonormal frame field ${\left\{e_i\right\}}_{i=1}^d$ on $R^d$, so that ${\left\{e_i\right\}}_{i=1}^m$ are tangent to M, and ${\left\{e_{\alpha }\right\}}_{\alpha =m+1}^d$ are normal to M. Meanwhile, we choose a fixed orthonormal basis ${\left\{E_A\right\}}_{A=1}^d$ on ${\mathbb{R}}^d$, and let

$$\begin{array}{c}\hfill W_A=\sum _{i=1}^m{W}_A^i{e}_i,W_A^i=\langle E_A,e_i\rangle ,W_A^{\alpha }=\langle E_A,e_{\alpha }\rangle ,\alpha =m+1,\cdots ,d.\end{array}$$

(74)

Therefore, we have $dF\left(W_A\right)\in \Gamma \left(F^{-1}TN\right)$. Using the components ${\mathcal{H}}_{ij}^{\alpha }$ of the second fundamental form of $M^m$ in $R^d$, we have

$$\begin{array}{ccc}\hfill & & \sum _{A=1}^d{W}_A^i{W}_A^j=\sum _{A=1}^d\langle E_A,e_i\rangle \langle E_A,e_j\rangle ={\delta }_{ij},\hfill \end{array}$$

(75)

$$\begin{array}{ccc}\hfill & & D_{e_i}{W}_A=\sum _{\alpha =m+1}^d\sum _{j=1}^m{\mathcal{H}}_{ij}^{\alpha }{W}_A^{\alpha }{e}_j,\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill & & {\tilde{D}}_{e_i}dF\left(W_A\right)=\sum _{\alpha =m+1}^d\sum _{k=1}^m{W}_A^{\alpha }{B}_{ik}^{\alpha }dF\left(e_k\right)+\sum _{k=1}^m{W}_A^k{\tilde{D}}_{e_i}dF\left(e_k\right).\hfill \end{array}$$

(77)

Substituting $V=W=dF\left(W_A\right)$ into (13), we obtain

$$\begin{array}{ccc}\hfill & & \sum _{A=1}^{d}I(dF\left(W_A\right),dF\left(W_A\right))\hfill \\ & =& (p-2){\int }_M{\left|dF\right|}^{p-4}\sum _{A,i,j}\langle {\tilde{D}}_{e_i}dF\left(W_A\right),dF\left(e_i\right)\rangle \langle {\tilde{D}}_{e_j}dF\left(W_A\right),dF\left(e_j\right)\rangle d{v}_g\hfill \\ & +& (q-2){\int }_M{\left|dF\right|}^{q-4}\sum _{A,i,j}\langle {\tilde{D}}_{e_i}dF\left(W_A\right),dF\left(e_i\right)\rangle \langle {\tilde{D}}_{e_j}dF\left(W_A\right),dF\left(e_j\right)\rangle d{v}_g\hfill \\ & +& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A,i}h({\tilde{D}}_{e_i}dF\left(W_A\right),{\tilde{D}}_{e_i}dF\left(V_A\right))d{v}_g\hfill \\ & +& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A,i}h(R^N(dF\left(V_A\right),dF\left(e_i\right))dF\left(V_A\right),dF\left(e_i\right))d{v}_g.\hfill \end{array}$$

(78)

Next, we can take an orthonormal basis ${\left\{e_i\right\}}_{i=1}^m$ such that $D_{e_i}{e}_j{|}_{P_0}=0$ for a fixed point $P_0$. In the following, we adopt the index following conventions: $i,j,k,l,\cdots \in \{1,\cdots ,m\},\alpha ,\beta ,\cdots \in \{m+1,\cdots ,d\},A\in \{1,\cdots ,d\}$. Because the matrix $h(dF\left(e_i\right),dF\left(e_j\right))$ is symmetric, we can take a local orthonormal basis ${\left\{e_i\right\}}_{i=1}^m$ such that $h(dF\left(e_i\right),dF\left(e_j\right))={\mu }_i^2{\delta }_{ij}$, $i,j=1,\cdots ,m.$ So we have ${\left|dF\right|}^2={\sum }_i^m{\mu }_i^2$. At $P_0$, using the symmetry of $DdF$, we obtain

$$\begin{array}{ccc}& & \sum _{A,i,j}\langle {\tilde{D}}_{e_i}dF\left(W_A\right),dF\left(e_i\right)\rangle \langle {\tilde{D}}_{e_j}dF\left(W_A\right),dF\left(e_j\right)\rangle \hfill \\ & =& \sum _{A,\alpha ,\beta }{W}_A^{\alpha }{W}_A^{\beta }\sum _{i,k,j,l}{\mathcal{H}}_{ik}^{\alpha }{\mathcal{H}}_{jl}^{\beta }h(dF\left(e_k\right),dF\left(e_i\right))h(dF\left(e_l\right),dF\left(e_j\right))\hfill \\ & & +\sum _A\sum _{i,k,j,l}{W}_A^k{W}_A^{l}h({\tilde{D}}_{e_i}dF\left(e_k\right),dF\left(e_i\right))h({\tilde{D}}_{e_j}dF\left(e_l\right),dF\left(e_j\right))\hfill \\ & =& \sum _{i,k,j,l,\alpha }{\mathcal{H}}_{ik}^{\alpha }{\mathcal{H}}_{jl}^{\alpha }h(dF\left(e_k\right),dF\left(e_i\right))h(dF\left(e_l\right),dF\left(e_j\right))\hfill \\ & & +\sum _{i,j,k}h(\left(D_{e_k}dF\right)\left(e_i\right),dF\left(e_i\right))h(\left(D_{e_k}dF\right)\left(e_j\right),dF\left(e_j\right)),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill & & \sum _{i,k,j,l,\alpha }{\mathcal{H}}_{ik}^{\alpha }{\mathcal{H}}_{jl}^{\alpha }h(dF\left(e_k\right),dF\left(e_i\right))h(dF\left(e_l\right),dF\left(e_j\right))\hfill \\ & =& \sum _{i,j,k,l}\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_j,e_l)\rangle {\mu }_k^2{\delta }_{ki}{\mu }_l^2{\delta }_{lj}\hfill \\ & =& \sum _{i,j}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle {\mu }_i^2{\mu }_j^2\hfill \\ & \le & \sum _{i,j}{\mu }_i^2{\mu }_j^2{\displaystyle \frac{1}{2}}\left[\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle +\langle \mathcal{H}(e_j,e_j),\mathcal{H}(e_j,e_j)\rangle \right]\hfill \\ & =& {\left|dF\right|}^2\sum _i{\mu }_i^2\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle .\hfill \end{array}$$

Therefore, the first and second terms on the right-hand side of Equation (78) are as follows:

$$\begin{array}{ccc}\hfill & & {\left|dF\right|}^{p-4}\sum _{A,i,j}\langle {\tilde{D}}_{e_i}dF\left(W_A\right),dF\left(e_i\right)\rangle \langle {\tilde{D}}_{e_j}dF\left(W_A\right),dF\left(e_j\right)\rangle \hfill \\ & \le & {\left|dF\right|}^{p-2}\sum _i{\mu }_i^2\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle \hfill \\ & & +{\left|dF\right|}^{p-4}\sum _{i,j,k}h(\left(D_{e_k}dF\right)\left(e_i\right),dF\left(e_i\right))h(\left(D_{e_k}dF\right)\left(e_j\right),dF\left(e_j\right))\hfill \end{array}$$

(79)

and

$$\begin{array}{ccc}\hfill & & {\left|dF\right|}^{q-4}\sum _{A,i,j}\langle {\tilde{D}}_{e_i}dF\left(W_A\right),dF\left(e_i\right)\rangle \langle {\tilde{D}}_{e_j}dF\left(W_A\right),dF\left(e_j\right)\rangle \hfill \\ & \le & {\left|dF\right|}^{q-2}\sum _i\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle {\mu }_i^2\hfill \\ & & +{\left|dF\right|}^{q-4}\sum _{i,j,k}h(\left(D_{e_k}dF\right)\left(e_i\right),dF\left(e_i\right))h(\left(D_{e_k}dF\right)\left(e_j\right),dF\left(e_j\right)).\hfill \end{array}$$

(80)

Next, we compute the third term of (78),

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,A}h({\tilde{D}}_{e_i}dF\left(W_A\right),{\tilde{D}}_{e_i}dF\left(W_A\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,A}h\left(\sum _{k,\alpha }{W}_A^{\alpha }{\mathcal{H}}_{ik}^{\alpha }dF\left(e_k\right)+\sum _k{W}_A^k{\tilde{D}}_{e_i}dF\left(e_k\right),\right.\hfill \\ & & \left.\sum _{l,\beta }{W}_A^{\beta }{\mathcal{H}}_{il}^{\beta }dF\left(e_l\right)+\sum _l{W}_A^l{\tilde{D}}_{e_i}dF\left(e_l\right)\right)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\left[\sum _{i,k,l,\alpha }{\mathcal{H}}_{ik}^{\alpha }{\mathcal{H}}_{il}^{\alpha }h(dF\left(e_k\right),dF\left(e_l\right))+\sum _{i,k}h(\left(D_{e_k}dF\right)\left(e_i\right),\left(D_{e_k}dF\right)\left(e_i\right))\right],\hfill \end{array}$$

(81)

where

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l,\alpha }{\mathcal{H}}_{ik}^{\alpha }{\mathcal{H}}_{il}^{\alpha }h(dF\left(e_k\right),dF\left(e_l\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l}\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_i,e_l)\rangle {\mu }_k^2{\delta }_{kl}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j}\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle {\mu }_i^2.\hfill \end{array}$$

(82)

Finally, we compute the last term of (78). Setting $\omega =dF\in \Gamma (T^{\ast }M\otimes F^{-1}TN)$ in Lemma 1, we have

$$\begin{array}{c}\hfill -\sum _{k=1}^m{R}^N(dF\left(Y\right),dF\left(e_k\right))dF\left(e_k\right)+dF\left(Ri{c}^{M^m}\left(Y\right)\right)=(\Delta dF)\left(Y\right)+\left(D^{2}dF\right)\left(Y\right).\end{array}$$

Therefore, we obtain

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A,i}h(R^N(dF\left(W_A\right),dF\left(e_i\right))dF\left(W_A\right),dF\left(e_i\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A}h((\Delta dF)\left(W_A\right),dF\left(W_A\right))\hfill \\ & & +\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A}h(\left(D^{2}dF\right)\left(W_A\right),dF\left(W_A\right))\hfill \\ & & -\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A}h(dF\left(Ri{c}^{M^m}\left(W_A\right)\right),dF\left(W_A\right)).\hfill \end{array}$$

(83)

Using ${\tau }_{p,q}\left(F\right)=-\delta \left(\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\right)=0$, (75), and (64), we see that each term on the right-hand side of Equation (83) is, respectively, given by

$$\begin{array}{ccc}\hfill & & {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _A\langle (\Delta dF)\left(W_A\right),dF\left(W_A\right)\rangle d{v}_g\hfill \\ & =& {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A,i,j}{W}_A^i{W}_A^j\langle (\Delta dF)\left(e_i\right),dF\left(e_j\right)\rangle d{v}_g\hfill \\ & =& {\int }_M〈\delta dF,\delta \left(\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)dF\right)〉d{v}_g=0\hfill \end{array}$$

(84)

and

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A}h(\left(D^{2}dF\right)\left(W_A\right),dF\left(W_A\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i}h(\left(D^{2}dF\right)\left(e_i\right),dF\left(e_i\right))\hfill \\ & =& \sum _k{e}_k\left\{\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i}h(D_{e_k}dF\left(e_i\right),dF\left(e_i\right))\right\}\hfill \\ & & -{(p-2)\left|dF\right|}^{p-4}\sum _{i,j,k}h(\left(D_{e_k}dF\right)\left(e_i\right),dF\left(e_i\right))h(\left(D_{e_k}dF\right)\left(e_j\right),dF\left(e_j\right))\hfill \\ & & -{(q-2)\left|dF\right|}^{q-4}\sum _{i,j,k}h(\left(D_{e_k}dF\right)\left(e_i\right),dF\left(e_i\right))h(\left(D_{e_k}dF\right)\left(e_j\right),dF\left(e_j\right))\hfill \\ & & -\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k}h(\left(D_{e_k}dF\right)\left(e_i\right),\left(D_{e_k}dF\right)\left(e_i\right))\hfill \end{array}$$

(85)

and

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{A}h(dF\left(Ri{c}^M\left(W_A\right)\right),dF\left(W_A\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,\alpha }\left[h\left(dF\left(\mathrm{trace}\left({\mathcal{A}}^{\alpha }\right){\mathcal{A}}^{\alpha }\left(e_i\right)\right),dF\left(e_i\right)\right)-h\left(dF\left({\mathcal{A}}^{\alpha }{\mathcal{A}}^{\alpha }\left(e_i\right)\right),dF\left(e_i\right)\right)\right],\hfill \end{array}$$

(86)

Using (63), the first and second terms on the right-hand side of Equation (86) are as follows:

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,\alpha }h\left(dF\left(\mathrm{trace}\left({\mathcal{A}}^{\alpha }\right){\mathcal{A}}^{\alpha }\left(e_i\right)\right),dF\left(e_i\right)\right)\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l,\alpha }\langle {\mathcal{A}}^{\alpha }\left(e_k\right),e_k\rangle \langle {\mathcal{A}}^{\alpha }\left(e_i\right),e_l\rangle h(dF\left(e_l\right),dF\left(e_i\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l}\langle \mathcal{H}(e_k,e_k),\mathcal{H}(e_i,e_l)\rangle {\mu }_l^2{\delta }_{li}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle {\mu }_i^2\hfill \end{array}$$

(87)

and

$$\begin{array}{ccc}\hfill & & \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,\alpha }h(dF\left({\mathcal{A}}^{\alpha }\left({\mathcal{A}}^{\alpha }\left(e_i\right)\right)\right),dF\left(e_i\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l,\alpha }\langle {\mathcal{A}}^{\alpha }\left(e_i\right),e_k\rangle \langle {\mathcal{A}}^{\alpha }\left(e_k\right),e_l\rangle h(dF\left(e_l\right),dF\left(e_i\right))\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,k,l}\langle \mathcal{H}(e_i,e_k),\mathcal{H}(e_l,e_k)\rangle {\mu }_l^2{\delta }_{li}\hfill \\ & =& \left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _{i,j}\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle {\mu }_i^2.\hfill \end{array}$$

(88)

By (78)–(88), we have

$$\begin{array}{ccc}& & \sum _{A}I(dF\left(W_A\right),dF\left(W_A\right))\hfill \\ & \le & {\int }_M\left({(p-2)\left|dF\right|}^{p-2}+(q-2){\left|dF\right|}^{q-2}\right)\sum _i{\mu }_i^2\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle d{v}_g\hfill \\ & & +{\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _i{\mu }_i^2\sum _j(2\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle )d{v}_g\hfill \\ & \le & {\int }_M\left({\left|dF\right|}^{p-2}+{\left|dF\right|}^{q-2}\right)\sum _i{\mu }_i^2\{max\{(p-2),(q-2)\}\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_i,e_i)\rangle \hfill \\ & & +\sum _j(2\langle \mathcal{H}(e_i,e_j),\mathcal{H}(e_i,e_j)\rangle -\langle \mathcal{H}(e_i,e_i),\mathcal{H}(e_j,e_j)\rangle )\}d{v}_g.\hfill \end{array}$$

Since M is a $(p,q)$-$SSU$ manifold, we see that if F is not constant, then

$$\sum _{A}I(dF\left(W_A\right),dF\left(W_A\right))<0.$$

This implies that F cannot be a stable $(p,q)$-harmonic map. Hence, a stable $(p,q)$-harmonic map must be constant. □

Remark 6. 

Take the source manifold as a unit sphere $S^m$ in ${\mathbb{R}}^{m+1}$. When $q=p$, Takeuchi [21] proved that each stable p-harmonic map from a unit sphere $S^m$ into any Riemannian manifold N is constant.

From Theorem 9 and Theorem 10, we have the following corollary:

Corollary 6. 

A stable $(p,q)$-harmonic map F from $S^m$ ($m>max\{p,q\}$) into a Riemannian manifold N, or from any compact Riemannian manifold into $S^n$ ($n>max\{p,q\}$), is necessarily constant.