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

## TL;DR

This paper investigates how eigenvalues of the Kohn Laplacian vary with pseudohermitian structures on CR manifolds, establishing continuity, differentiability, and conditions for critical structures, with explicit examples provided.

## Contribution

It introduces a framework for analyzing eigenvalue functionals on CR manifolds, including continuity, differentiability, and criteria for critical structures, with explicit examples.

## Key findings

- Eigenvalues are continuous functionals on the space of pseudohermitian structures.
- Eigenvalue functionals are (one-sided) differentiable along analytic deformations.
- Explicit examples of critical pseudohermitian structures are constructed.

## 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.