Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds

Amine Aribi, Duong Ngoc Son

Abstract

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures $\mathcal{P}_{+}$. We show that the functionals are continuous with respect to a natural topology on $\mathcal{P}_{+}$. Using an adaptation of the standard Kato--Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical pseudohermitian structures, in a generalized sense, for them. We give a necessary (also sufficient in some situations) condition for a pseudohermitian structure to be critical. Finally, we present explicit examples of critical pseudohermitian structures on both homogeneous and non-homogeneous CR manifolds.

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds

Abstract

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures . We show that the functionals are continuous with respect to a natural topology on . Using an adaptation of the standard Kato--Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical pseudohermitian structures, in a generalized sense, for them. We give a necessary (also sufficient in some situations) condition for a pseudohermitian structure to be critical. Finally, we present explicit examples of critical pseudohermitian structures on both homogeneous and non-homogeneous CR manifolds.

Paper Structure

This paper contains 11 sections, 19 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Continuity of the eigenvalue functionals
  3. The Kohn Laplacian on pseudohermitian manifolds
  4. The Max-mini principle
  5. Proof of Theorem \ref{['thm:main']} and Corollary \ref{['cor:semicont']}
  6. Derivatives of the eigenvalue functionals
  7. Proofs of Theorem \ref{['thm:main2']} and Corollary \ref{['cor:mul']}
  8. Proof of Theorem \ref{['thm:main2']}
  9. A refinement
  10. Eigenvalue ratio functionals
  11. Examples

Key Result

Theorem \oldthetheorem

Let $M^{2n+1}$ be a compact strictly pseudoconvex embeddable CR manifold. Suppose that $\theta$ and $\hat{\theta} = e^{u}\theta$ are two pseudohermitian structures on $M$. For $\delta>0$ and $\delta'>0$, if $\sup_M |u| < \delta$ and $\sup_M |\bar{\partial}_b u|_{\theta} < \delta'$, then In particular, the map $\theta \mapsto \lambda_k(\theta)$ is locally Lipschitz continuous on $(\mathcal{P}_{+},

Theorems & Definitions (37)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem: Max-mini Principle
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 27 more