Existence and Stability of α−Harmonic Maps

Abstract

In this paper, we first study the α − energy functional, Euler-Lagrange operator, and α -stress-energy tensor. Second, it is shown that the critical points of the α − energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α − 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 α − harmonic maps are minimal submanifolds. Then, the stability of any α − harmonic map on Riemannian manifold with nonpositive curvature is studied. Finally, the instability of α − harmonic maps from a compact manifold to a standard unit sphere is investigated.