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

Francesco Bei, Giuseppe Pipoli

TL;DR

This work proves finiteness and vanishing theorems for $L^2$-harmonic forms and spinors on complete, stable minimal (and strongly stable CMC) hypersurfaces, under positive or controlled curvature assumptions on the ambient manifold. A general Schrödinger/Dirac framework with weighted Poincaré inequalities yields essential self-adjointness and finite-dimensionality of $L^2$ kernels, with constant-length properties for $L^2$-harmonic spinors and dimension bounds. These analytic results translate into strong geometric consequences: constrained nullity, conformal Laplacian behavior, scalar-flat/positive metrics in conformal classes, and extensive vanishing results for $L^2$-forms under curvature and second fundamental form bounds. The authors illustrate the theory on Euclidean spaces, manifolds with nonnegative curvature operator, and Berger spheres, providing explicit thresholds for spinor and form vanishings, and extend the approach to strongly stable CMC hypersurfaces, highlighting the role of the scalar and bi-Ricci curvature in controlling $L^2$-cohomology and spinors.

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.

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

TL;DR

This work proves finiteness and vanishing theorems for -harmonic forms and spinors on complete, stable minimal (and strongly stable CMC) hypersurfaces, under positive or controlled curvature assumptions on the ambient manifold. A general Schrödinger/Dirac framework with weighted Poincaré inequalities yields essential self-adjointness and finite-dimensionality of kernels, with constant-length properties for -harmonic spinors and dimension bounds. These analytic results translate into strong geometric consequences: constrained nullity, conformal Laplacian behavior, scalar-flat/positive metrics in conformal classes, and extensive vanishing results for -forms under curvature and second fundamental form bounds. The authors illustrate the theory on Euclidean spaces, manifolds with nonnegative curvature operator, and Berger spheres, providing explicit thresholds for spinor and form vanishings, and extend the approach to strongly stable CMC hypersurfaces, highlighting the role of the scalar and bi-Ricci curvature in controlling -cohomology and spinors.

Abstract

Let be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of -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 or carries no non-trivial -harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.
Paper Structure (7 sections, 27 theorems, 131 equations, 1 figure)

This paper contains 7 sections, 27 theorems, 131 equations, 1 figure.

Table of Contents

  1. Finiteness and vanishing theorems for Schrödinger and Dirac operators
  2. Applications to stable minimal hypersurfaces
  3. Examples
  4. Euclidean space
  5. Spaces with nonnegative curvature operator
  6. Berger spheres
  7. Strongly stable constant mean curvature hypersurfaces

Key Result

Theorem 1

Let $(M,g)$ be a Riemannian manifold with $s_g\geq 0$. Let $N$ be a spinnable manifold with $\dim(N)+1=\dim(M)$ such that there exists a two-sided, stable minimal immersion with $(N,g_N)$ complete. Let $P_{\mathrm{Spin(n)}}(N)\rightarrow N$ be an arbitrarily fixed Riemannian spin structure on $N$ and let $(\Sigma N,\tau)\rightarrow N$ and $\eth$ be the corresponding spinor bundle and spin-Dirac o

Figures (1)

  • Figure 1: Comparison of different notion of curvature for the Berger spheres $(\mathbb S^{2n+1},g_{\delta})$ for $n>6$. When $n<6$ (resp. $n=6$) the order of the special values of $\delta$ is the same except that $\frac{2n+2}{2n+1}$ and $\frac{4(2n^2+n+6)}{8n^2+n+18}$ are interchanged (resp. coincide) .

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Theorem 1.1
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • proof
  • Corollary 1.1
  • proof
  • ...and 49 more