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

Abstract

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.