Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866911853404225536 |
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| author | Aribi, Amine Son, Duong Ngoc |
| author_facet | Aribi, Amine Son, Duong Ngoc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_05881 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds Aribi, Amine Son, Duong Ngoc Complex Variables Spectral Theory 32V20, 32W10, 58C40 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. |
| title | Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds |
| topic | Complex Variables Spectral Theory 32V20, 32W10, 58C40 |
| url | https://arxiv.org/abs/1901.05881 |