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On the generalized of $p$-biharmonic and bi-$p$-harmonic maps

Fethi Latti, Ahmed Mohammed Cherif

Abstract

In this note, we extend the definition of $p$-biharmonic and bi-$p$-harmonic maps between two Riemannian manifolds and explore some of their properties.

On the generalized of $p$-biharmonic and bi-$p$-harmonic maps

Abstract

In this note, we extend the definition of -biharmonic and bi--harmonic maps between two Riemannian manifolds and explore some of their properties.
Paper Structure (3 sections, 7 theorems, 44 equations)

This paper contains 3 sections, 7 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. First variation of the $(p,q)$-energy functional
  3. Liouville-type theorems for $(p,q)$-harmonic maps

Key Result

Theorem 1

Let $\varphi$ be a smooth map from Riemannian manifold $(M,g)$ to Riemannian manifold $(N,h)$, and $\{\varphi_{t}\}_{t\in(-\epsilon,\epsilon)}$ a smooth variation of $\varphi$ to support in compact domain $D\subset M$. Then where $\tau_{p,q}(\varphi)$ is the $(p,q)$-tension field of $\varphi$ given by and $v=\frac{d\varphi_{t}}{dt}|_{t=0}$ denotes the variation vector field of $\{\varphi_{t}\}_{

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Remark 5
  • Example 6
  • Corollary 7
  • Example 8
  • Remark 9
  • ...and 5 more