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 ${\mathcal{P}}_{+}$. 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.