  We survey some $L^{p}$-vanishing results for solutions of Bochner or Simons
type equations with refined Kato inequalities, under spectral assumptions on
the relevant Schr\"{o}dinger operators. New aspects are included in the
picture. In particular, an abstract version of a structure theorem for stable
minimal hypersurfaces of finite total curvature is observed. Further geometric
applications are discussed.
