Existence and Stability ofα−Harmonic Maps

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 the$\alpha -$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 on Riemannian manifold with nonpositive curvature is studied. Finally, the instability of$\alpha -$harmonic maps from a compact manifold to a standard unit sphere is investigated.