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

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 Emery 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 Emery 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. (C) 2018 Elsevier Inc. All rights reserved.