L2$L^2$ ‐harmonic forms and spinors on stable minimal hypersurfaces

Let f:N→(M,g)$f:N\rightarrow (M,g)$ be a two‐sided, complete, stable, minimal, immersed hypersurface. In this paper, we establish various vanishing theorems for the space of L2$L^2$ ‐harmonic forms and spinors (when M$M$ 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 Rm$\mathbb {R}^m$ or Sm$\mathbb {S}^m$ carries no non‐trivial L2$L^2$ ‐harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.