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

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.

💡 Research Summary

The paper investigates the existence of L²‑harmonic differential forms and L²‑harmonic spinors on complete, two‑sided, stable minimal hypersurfaces immersed in a Riemannian manifold (M,g). The authors develop general vanishing and finiteness results for the L²‑kernel of Schrödinger operators and Dirac operators on complete manifolds under a weighted Poincaré inequality. These analytic results are then applied to the geometry of stable minimal hypersurfaces.

In the first part, they prove that if a formally self‑adjoint Schrödinger operator P=∇*∇+L satisfies L+q≥0 and (Δ−q)≥0 in the sense of quadratic forms, then P is essentially self‑adjoint on C_c^∞ sections, its L²‑kernel is finite‑dimensional and bounded by the rank of the bundle, and any L²‑section in the kernel cannot vanish on an open set. When q≥0, any L²‑section in the kernel must have constant length. Analogous statements are obtained for the Dirac operator.

The second part translates these analytic statements into geometric vanishing theorems. If the ambient manifold has non‑negative Ricci curvature, the nullity of a two‑sided stable minimal hypersurface is either 0 or 1. If the ambient scalar curvature is non‑negative, the conformal Laplacian on the hypersurface is essentially self‑adjoint with finite‑dimensional L²‑kernel, leading to the existence of conformally related metrics with prescribed scalar curvature.

The main spinorial result (Theorem 0.1) states that for a stable minimal hypersurface N in a Riemannian manifold (M,g) with non‑negative scalar curvature, any L²‑harmonic spinor on N has constant length and the dimension of the space of L²‑harmonic spinors is at most 2⌊n/2⌋ (or 2⌊(n−1)/2⌋ when n is odd). If vol(N)=∞, then N carries no non‑trivial L²‑harmonic spinors. Consequently, the “Spinorial Tanno problem’’ is completely solved for oriented complete stable minimal hypersurfaces in ℝ^m and S^m.

For differential forms, the authors introduce a curvature condition involving the second fundamental form A and the principal curvature sums K_α: |A|²−K_α²≥0 for every multi‑index α of size p. Combined with curvature bounds on the ambient manifold (non‑negative curvature operator, or sectional curvature lying in a small interval