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

TL;DR

This paper proves vanishing theorems for $L^2$-harmonic forms and spinors on stable minimal hypersurfaces under positive curvature conditions, extending previous results to higher dimensions and more general ambient spaces.

Contribution

It generalizes vanishing theorems for $L^2$-harmonic forms and spinors to higher dimensions and broader ambient manifolds, including stable minimal and constant mean curvature hypersurfaces.

Findings

Stable minimal hypersurfaces in Euclidean and spherical spaces have no non-trivial $L^2$-harmonic spinors.

Vanishing theorems extend to higher dimensions and more general ambient Riemannian manifolds.

Results apply to strongly stable constant mean curvature hypersurfaces.

Abstract

Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) 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 \cite{Tanno} and Zhu \cite{Zhu}. In the setting of spin manifolds our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $\mathbb{R}^m$ or $\mathbb{S}^m$ carries no non-trivial $L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.