Some isolation results for f-harmonic maps on weighted Riemannian manifolds with boundary

Abstract

In this paper, we study f -harmonic (respectively, ( p , f ) -harmonic) maps on a compact weighted Riemannian manifold of nonempty boundary and of positive Bakry–Émery Ricci curvature. We first establish a Bochner–Reilly formula for such maps and deduce therefrom some immediate isolation results. In addition, using a weighted Sobolev inequality obtained in [12] , we prove that, under some energy level depending on the Bakry–Émery curvature of the initial manifold and on the sectional curvature of the final one, the only f -harmonic (respectively, ( p , f ) -harmonic) maps are the constant ones. A gap property of the energy density of f -harmonic maps from a weighted Riemannian manifold to a unit sphere is also obtained.