Article
Kyungpook Mathematical Journal 2021; 61(1): 169-179
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
On the Generalized of p-harmonic and f-harmonic Maps
Embarka Remli and Ahmed Mohammed Cherif*
Mascara University, Faculty of Exact Sciences, Department of Mathematics, 29000, Algeria
e-mail : ambarka.ramli@univ-mascara.dz and a.mohammedcherif@univ-mascara.dz
Received: February 15, 2020; Accepted: August 18, 2020
Abstract
In this paper, we extend the definition of p-harmonic maps between two Riemannian manifolds. We prove a Liouville type theorem for generalized p-harmonic maps. We present some new properties for the generalized stress p-energy tensor. We also prove that every generalized p-harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a homothetic vector field satisfying some condition is constant.
Keywords: p-harmonic maps, f-harmonic maps, Liouville type theorem, stress energy tensor.
1. (p,f) -harmonic Maps
If
where
(for more details on the concept of
where
Then
Let
where
We call
Theorem 1.1.
(The first variation of the
where
and
First, note that
Substituting the last formula in (1.8), and using
Let
So that, the divergence of ω at
By the equations (1.9), (1.10), we get
The Theorem 1.1 follows from (1.11), and the divergence Theorem.
From Theorem 1.1, we deduce:
Theorem 1.2.
Let
Example 1.3.
According to Theorem 1.2, the inversion map
is
Remark 1.4.
In particular, we note that every harmonic map with constant energy density
2. A Liouville Type Theorem for (p,f) -harmonic Maps
Liouville type theorems for harmonic maps between complete smooth Riemannian manifolds have been done by many authors. Liu [10] proved the Liouville-type theorem for
The purpose of this section is to provide a proof of the Liouville type theorem for
Theorem 2.1.
Let
Then
We will need the following lemma to prove the Theorem 2.1.
Lemma 2.2.
([5, 16]) Let
Here
Let
where
Since the map
Let
where
Note that by the
By using the Bochner formula (2.1), with
We set
By using the following formula
and inequality (2.3), we have the following
From the Kato's inequality
Let
By the Young inequality, we have
and the following inequality
for any
By using the divergence Theorem, with
Choose the smooth cut-off
Since
Thus, if
But
From Theorem 2.1, we deduce:
Corollary 2.2.
([13, 14]) Let
3. Stress (p,f) -energy Tensor
Let
Let
where
Theorem 3.1.
Under the notation above we have the following
where
From Theorem 3.1, we deduce:
Theorem 3.2.
A non-constant smooth map
Theorem 3.3.
Let
By the definitions of gradient and
The Theorem 3.3 follows from (3.3), and the definition of
4. Homothetic Vector Fields and (p,f) -harmonic Maps
A vector field
If
In the seminal work [11], where we proved that, if
Theorem 4.1.
Let
Then
and let
By equation (4.2), and
Since ξ is a homothetic vector field with homothetic constant
is equivalent to the following equation
By the Young's inequality, we have
for all
from 4.3, 4.4, we deduce the following inequality
We assume that
From 4.6, and the divergence Theorem, we have
Now, consider the cut-off smooth function
since
Consequently,
Corollary 4.2.
Let
Example 4.3.
Let
is harmonic, where
is
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