Stability and quantum phenomenen and Liouville theorems of $p$-harmonic maps with potential

Abstract

In this paper, we discuss the stability and the pointwise gap phenomenen of $p$-harmonic maps with potential. Stability theorems of $p$-$H$-harmonic maps from or into general submanifolds of the shpere and the Euclidean space are established, and Sealey's quantum theorem is extended. We also discuss the conservation law and the Liouville theorems of $p$-$H$-harmonic maps. As a consequence of our stability theorem, we not only generalize Leung's stability theorem to rather general case, but also improve it by replacing the sectional curvature bound by a Ricci curvature bound. In order to discuss the gap property of $p$-harmonic maps, we establish a Bochner-typed formula which is used by some authors in a uncorrect form.