Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the generalization of biharmonic hypersurfaces and biharmonic curves

Moustafa Tadj, Ahmed Mohammed Cherif, Fethi Latti

Abstract

In this work, we extend the concepts of $p$-biharmonic maps and $p$-biharmonic hypersurfaces to provide a broader characterization of $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in Riemannian manifolds, including Einstein spaces. Moreover, we present new explicit examples of proper $(p,q)$-harmonic hypersurfaces and $(p,q)$-harmonic curves in space forms.

On the generalization of biharmonic hypersurfaces and biharmonic curves

Abstract

In this work, we extend the concepts of -biharmonic maps and -biharmonic hypersurfaces to provide a broader characterization of -harmonic hypersurfaces and -harmonic curves in Riemannian manifolds, including Einstein spaces. Moreover, we present new explicit examples of proper -harmonic hypersurfaces and -harmonic curves in space forms.
Paper Structure (4 sections, 8 theorems, 63 equations)

This paper contains 4 sections, 8 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The $(p,q)$-Harmonic maps
  3. The $(p,q)$-harmonic hypersurfaces
  4. $(p,q)$-harmonic curves in $3$-dimensional space form

Key Result

Theorem 1

Let $\varphi$ be a smooth map from a Riemannian manifold $(M,g)$ to a Riemannian manifold $(N,h)$, and let $\{\varphi_t\}_{t\in(-\varepsilon,\varepsilon)}$ be a smooth variation of $\varphi$ with compact support in a 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 $

Theorems & Definitions (17)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Proposition 4
  • Corollary 5
  • proof
  • Corollary 6
  • proof
  • ...and 7 more