The Morse Index of Sacks–Uhlenbeck α-Harmonic Maps for Riemannian Manifolds

In this paper, first we prove a nonexistence theorem for α-harmonic mappings between Riemannian manifolds. Second, the instability of nonconstant α-harmonic maps is studied with regard to the Ricci curvature criterion of their codomain. Then, we estimate the Morse index for measuring the degree of instability of some particular α-harmonic maps. Furthermore, the notion of α-stable manifolds and its applications are considered. Finally, we investigate the α-stability of any compact Riemannian manifolds admitting a nonisometric conformal vector field and any Einstein Riemannian manifold under certain assumptions on the smallest positive eigenvalue of its Laplacian operator on functions.