$L^2$‐harmonic forms and spinors on stable minimal hypersurfaces

Let (Formula presented.) be a two-sided, complete, stable, minimal, immersed hypersurface. In this paper, we establish various vanishing theorems for the space of (Formula presented.) -harmonic forms and spinors (when (Formula presented.) is additionally spin) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno [J. Math. Soc. Japan 48 (1996), no. 4, 761–768] and Zhu [Nonlinear Anal. 75 (2012), no. 13, 5039–5043]. In the setting of spin manifolds, our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of (Formula presented.) or (Formula presented.) carries no non-trivial (Formula presented.) -harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.