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CR Paneitz operator on non-embeddable CR manifolds

Yuya Takeuchi

TL;DR

The paper analyzes the CR Paneitz operator $P$ on non-embeddable CR 3-manifolds, proving essential self-adjointness and that $ ext{Spec}\,P$ is discrete except at $0$, with nonzero eigenvalues of finite multiplicity and smooth eigenfunctions. It develops the Heisenberg calculus toolkit, including approximate Szegő projections and parametrix constructions, to establish the spectral properties for $P$ and its domain. Focusing on the Rossi sphere, the authors use spherical harmonics to show that the CR Paneitz operator has infinitely many negative eigenvalues, highlighting a stark contrast with embeddable CR geometry. The results illuminate how non-embeddability influences spectral behavior and raise questions about closed-range phenomena and broader applicability to CR Yamabe-type problems.

Abstract

The CR Paneitz operator is closely related to some important problems in CR geometry. In this paper, we consider this operator on a non-embeddable CR manifold. This operator is essentially self-adjoint and its spectrum is discrete except zero. Moreover, the eigenspace corresponding to each non-zero eigenvalue is a finite dimensional subspace of the space of smooth functions. Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an example of non-embeddable CR manifolds, has infinitely many negative eigenvalues, which is significantly different from the embeddable case.

CR Paneitz operator on non-embeddable CR manifolds

TL;DR

The paper analyzes the CR Paneitz operator on non-embeddable CR 3-manifolds, proving essential self-adjointness and that is discrete except at , with nonzero eigenvalues of finite multiplicity and smooth eigenfunctions. It develops the Heisenberg calculus toolkit, including approximate Szegő projections and parametrix constructions, to establish the spectral properties for and its domain. Focusing on the Rossi sphere, the authors use spherical harmonics to show that the CR Paneitz operator has infinitely many negative eigenvalues, highlighting a stark contrast with embeddable CR geometry. The results illuminate how non-embeddability influences spectral behavior and raise questions about closed-range phenomena and broader applicability to CR Yamabe-type problems.

Abstract

The CR Paneitz operator is closely related to some important problems in CR geometry. In this paper, we consider this operator on a non-embeddable CR manifold. This operator is essentially self-adjoint and its spectrum is discrete except zero. Moreover, the eigenspace corresponding to each non-zero eigenvalue is a finite dimensional subspace of the space of smooth functions. Furthermore, we show that the CR Paneitz operator on the Rossi sphere, an example of non-embeddable CR manifolds, has infinitely many negative eigenvalues, which is significantly different from the embeddable case.
Paper Structure (9 sections, 22 theorems, 98 equations)

This paper contains 9 sections, 22 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. CR manifolds
  3. Heisenberg calculus
  4. Proof of \ref{['thm:ess-self-adj-of-CR-Paneitz', 'thm:spectrum-of-CR-Paneitz']}
  5. CR Paneitz operator on the Rossi sphere
  6. Definition of the Rossi sphere
  7. Spherical harmonics
  8. Negative eigenvalues of the CR Paneitz operator
  9. Concluding remarks

Key Result

Theorem 1.1

The CR Paneitz operator $P$ is essentially self-adjoint; equivalently, the maximal closed extension of $P$ is self-adjoint.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 3.1: Ponge2008-Book*Propositions 3.2.6 and 3.2.9
  • Lemma 3.2: Takeuchi2023-GJMS*Lemma 4.2
  • Proposition 3.3: Ponge2008-Book*Proposition 3.3.1
  • Proposition 3.4: Ponge2008-Book*Propositions 5.5.8 and Takeuchi2023-GJMS*Proposition 4.6
  • Theorem 4.1: Beals-Greiner1988*Proposition 25.4 and Corollaries 25.64 and 25.67
  • Lemma 4.2
  • proof
  • ...and 29 more