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Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds

Georges Habib, Felipe Leitner

TL;DR

This work derives a new eigenvalue bound for the Kohn-Dirac operator $D_\theta$ on closed CR spin manifolds, requiring only a lower bound on the Webster scalar curvature ${\rm Scal}^W$ and relating extremal cases to CR twistor spinors. The authors decompose the CR spinor bundle into $\Sigma_r(H(M))$ and introduce CR twistor operators $T_r$ to obtain lower bounds for $D_\theta^2$ on each component, with sharp equality characterizations involving constant ${\rm Scal}^W$, torsion vanishing, and CR twistor spinors. They further classify manifolds supporting such spinors by showing the Webster Ricci tensor ${\rm Ric}^W$ has at most two eigenvalues, and they provide explicit geometric constructions (pseudo-Einstein and middle-slot two-eigenvalue cases) illustrating the theory in practice. The results connect CR spin geometry with spectral properties of $D_\theta$, the structure of ${\rm Ric}^W$, and Fefferman-type constructions, offering concrete examples and a framework for CR manifolds with special spinor fields.

Abstract

In this paper, we establish a new eigenvalue estimate for the Kohn-Dirac operator on a compact CR manifold. The equality case of this estimate is characterized by the existence of a CR twistor spinor. We then classify CR manifolds carrying such spinors by showing that the Webster Ricci tensor has at most two eigenvalues. In this context, we construct several examples.

Eigenvalue estimates of the Kohn-Dirac operator on CR manifolds

TL;DR

This work derives a new eigenvalue bound for the Kohn-Dirac operator on closed CR spin manifolds, requiring only a lower bound on the Webster scalar curvature and relating extremal cases to CR twistor spinors. The authors decompose the CR spinor bundle into and introduce CR twistor operators to obtain lower bounds for on each component, with sharp equality characterizations involving constant , torsion vanishing, and CR twistor spinors. They further classify manifolds supporting such spinors by showing the Webster Ricci tensor has at most two eigenvalues, and they provide explicit geometric constructions (pseudo-Einstein and middle-slot two-eigenvalue cases) illustrating the theory in practice. The results connect CR spin geometry with spectral properties of , the structure of , and Fefferman-type constructions, offering concrete examples and a framework for CR manifolds with special spinor fields.

Abstract

In this paper, we establish a new eigenvalue estimate for the Kohn-Dirac operator on a compact CR manifold. The equality case of this estimate is characterized by the existence of a CR twistor spinor. We then classify CR manifolds carrying such spinors by showing that the Webster Ricci tensor has at most two eigenvalues. In this context, we construct several examples.

Paper Structure

This paper contains 9 sections, 10 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Review on CR manifolds
  3. Spin structure on CR manifolds
  4. Eigenvalue estimates
  5. CR twistor spinors in $\Sigma_r(H(M))$, when $r\neq\frac{m}{2}$
  6. CR twistor spinors in $\Sigma_{\frac{m}{2}}(H(M))$
  7. Examples
  8. Pseudo-Einstein CR manifold
  9. Two eigenvalues of the Webster Ricci tensor in the middle slot

Key Result

Lemma 2.2

On the space of horizontal differential forms, we have where $\tau^{[k]}$ is the canonical extension of the symmetric endomorphism $\tau$ given for any $\omega\in \Omega^k(H(M))$ by for all $X_1,\ldots,X_k\in H(M)$.

Theorems & Definitions (23)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 13 more