Existence and Stability of $\alpha-$ harmonic Maps

TL;DR

This paper investigates the properties, stability, and construction of $ ext{alpha}$-harmonic maps between Riemannian manifolds, linking them to harmonic maps and analyzing their stability under various curvature conditions.

Contribution

It introduces the $ ext{alpha}$-energy functional, relates $ ext{alpha}$-harmonic maps to harmonic maps via conformal deformation, and studies their stability and minimality conditions.

Findings

Critical points of $ ext{alpha}$-energy relate to harmonic maps.

Conditions for fibers of conformal $ ext{alpha}$-harmonic maps to be minimal.

Stability of $ ext{alpha}$-harmonic maps from non-positively curved manifolds.

Abstract

In this paper, we first study the $\alpha-$energy functional, Euler-Lagrange operator and $\alpha$-stress energy tensor. Second, it is shown that the critical points of $\alpha-$ energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an $\alpha-$harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal $\alpha-$ harmonic maps are minimal submanifolds. Then, the stability of any $\alpha-$harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of $\alpha-$harmonic maps from a compact manifold to a standard unit sphere is investigated.