Nonexistence of proper $p$-biharmonic maps and Liouville type theorems I: case of $p\geq2$

TL;DR

This paper proves the nonexistence of proper p-biharmonic maps for p ≥ 2 and establishes Liouville type theorems for maps from Euclidean space under integral conditions, extending previous results.

Contribution

It demonstrates the nonexistence of proper p-biharmonic maps for p ≥ 2 and extends Liouville type theorems to broader conditions, advancing understanding of p-biharmonic map behavior.

Findings

Proper p-biharmonic maps do not exist for p ≥ 2.

Liouville type theorems hold for maps from Euclidean space under certain integral conditions.

Extension of previous results by Baird, Fardoun, and Ouakkas.

Abstract

Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and $(N, h)$. The $p$-bienergy of $u$ is defined by $E_p(u)=\int_M|\tau(u)|^pd\nu_g$, where $\tau(u)$ is the tension field of $u$ and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. In this paper we will prove nonexistence result of proper $p$-biharmonic maps when $p\geq2$. In particular when $M=\mathbb{R}^m$, we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas \cite{BFO}.