The Stability of $\alpha-$ Harmonic Maps with Physical Applications

TL;DR

This paper explores the stability and existence of $oldsymbol{ extalpha}$-harmonic maps, establishing non-existence results, connections to harmonic maps, and analyzing stability conditions with applications to physical models.

Contribution

It introduces new non-existence theorems for $oldsymbol{ extalpha}$-harmonic maps and investigates their stability, linking geometric analysis with physical applications.

Findings

Non-existence theorem for $oldsymbol{ extalpha}$-harmonic mappings

Connection between $oldsymbol{ extalpha}$-harmonic and harmonic maps via conformal deformation

Conditions for $oldsymbol{ extalpha}$-stability on Riemannian manifolds

Abstract

The first result in this study is a non-existence theorem for $\alpha-$harmonic mappings. Additionally, a direct connection between the $\alpha-$ harmonic and harmonic maps is made possible via conformal deformation. Second, the instability of non-constant $\alpha$-harmonic maps is investigated with regard to the target manifold's Ricci curvature requirements. Next, the concept of $\alpha-$stable manifolds and their physical applications are explored. Finally, it is investigated the $\alpha-$stability of compact Riemannian manifolds that admit a non-isometric conformal vector field as well as the Einstein Riemannian manifolds under certain assumption on the smallest positive eigenvalue of its Laplacian operator on functions.