CR Geometry, the ∂̄_b-Problem, and the Lewy Extension¶
Job: develop the CR manifolds / Levi form / pseudoconvexity, the tangential \(\bar\partial_b\) / Kohn–Rossi Laplacian / Tanaka–Webster connection, the Lewy extension / Hansson–Lewy non-solvability, the CR embeddability (Boutet de Monvel / Rossi), and the Heisenberg group / Folland–Stein operators — with the Heisenberg / Folland–Stein \(\bar\partial_b\) / Lewy non-solvability as the worked example.
Core¶
CR manifolds and the CR structure¶
A CR manifold (Cauchy–Riemann manifold) \(M^{2n-1}\) (a real hypersurface in \(\mathbb{C}^n\), or an abstract CR manifold) is a real manifold with a complex subbundle \(T^{0,1}M\subset\mathbb{C}\otimes TM\) (the \((0,1)\)-vectors, the Cauchy–Riemann directions) of rank \(n-1\), satisfying the integrability \([T^{0,1},T^{0,1}]\subset T^{0,1}\).
The Levi form \(\mathcal{L}(\xi,X)=\langle\bar\partial\theta(\bar X,X),\xi\rangle\) (the hermitian form on \(T^{0,1}\), the pseudoconvexity of the CR structure: the Levi form positive \(\Leftrightarrow\) the strictly pseudoconvex / the boundary of a strictly pseudoconvex domain). The Levi-flat CR manifolds (the Levi form \(=0\)) are the foliated / the real-complex (the codimension-1 Levi foliation).
| CR structure | Description |
|---|---|
| CR manifold \(M^{2n-1}\) | a real manifold with \(T^{0,1}M\subset\mathbb{C}\otimes TM\) of rank \(n-1\), integrable |
| Levi form \(\mathcal{L}\) | the hermitian form on \(T^{0,1}\), the pseudoconvexity (positive \(\Leftrightarrow\) strictly pseudoconvex) |
| Levi-flat | \(\mathcal{L}=0\) (the codimension-1 Levi foliation, the foliated CR) |
| Strictly pseudoconvex CR | \(\mathcal{L}>0\) (the boundary of a strictly pseudoconvex domain) |
The \(\bar\partial_b\)-problem and the Kohn–Rossi Laplacian¶
| \(\bar\partial_b\) data | Description |
|---|---|
| Tangential \(\bar\partial_b\) | the restriction of \(\bar\partial\) to the CR directions (the boundary operator on \(M=\partial\Omega\)) |
| Kohn–Rossi Laplacian \(\Box_b=\bar\partial_b^*\bar\partial_b+\bar\partial_b\bar\partial_b^*\) | the analog of the \(\bar\partial\)-Neumann Laplacian, on the boundary CR forms |
| Kohn–Rossi cohomology \(H^q_b(M)\) | the \(\bar\partial_b\)-cohomology (the CR-cohomology) |
| Tanaka–Webster connection | the canonical connection on a strictly pseudoconvex CR (the pseudo-hermitian geometry) |
The Lewy extension theorem and the Hansson–Lewy non-solvability¶
Theorem (Lewy extension). For a strictly pseudoconvex CR hypersurface \(M\subset\mathbb{C}^n\), a CR function on \(M\) (a function annihilated by \(\bar\partial_b\)) extends holomorphically to one side of \(M\) (the holomorphic extension to the pseudoconvex side, the Hansson–Wermer for the embedding / the CR extension).
The Lewy example / the non-solvability. The Lewy operator \(L=\partial_x+iy\partial_z\) (the Hansson–Lewy counterexample, the non-solvability of \(Lu=f\) for some smooth \(f\) — the first example of a non-solvable PDE / the non-hypoellipticity, the non-solvability of the \(\bar\partial_b\) on a non-pseudoconvex CR).
| Lewy extension / non-solvability | Description |
|---|---|
| Lewy extension | the CR function on a strictly pseudoconvex \(M\) extends holomorphically to the pseudoconvex side |
| Hansson–Lewy non-solvability | the Lewy operator \(L=\partial_x+iy\partial_z\) not solvable for some smooth \(f\) (the non-pseudoconvex CR) |
| CR extension vs PDE solvability | the strict pseudoconvexity gives the solvability, the non-pseudoconvex the non-solvability |
The CR embeddability and the strict pseudoconvexity¶
Theorem (Boutet de Monvel CR embedding). A compact strictly pseudoconvex CR manifold of dimension \(\ge 5\) embeds as a real hypersurface in \(\mathbb{C}^n\) (the CR embedding).
| CR embedding | Description |
|---|---|
| Boutet de Monvel | the compact strictly pseudoconvex CR of dim \(\ge 5\) embeds in \(\mathbb{C}^n\) |
| Rossi non-embeddability | the CR structures that do not embed (the non-strict / the exceptional CR) |
| Szegő kernel / CR | the Kohn–Folland–Rossi / the boundary behavior of the Bergman / Szegő kernel on the strictly pseudoconvex CR |
The Heisenberg group and the CR geometry¶
| Heisenberg / Folland–Stein | Description |
|---|---|
| Heisenberg group \(\mathbb{H}^{2n+1}\) | the boundary of the Siegel upper half-space, the model strictly pseudoconvex CR, the nilpotent Lie group |
| Folland–Stein operators | the sub-Laplacian / the Kohn Laplacian on \(\mathbb{H}\) (the fundamental solution, the subelliptic estimates) |
| CR automorphisms | the Korányi–Reimann (the contact transformations preserving the CR structure) |
See it¶
Heisenberg group as the model strictly pseudoconvex CR and the Folland–Stein \(\bar\partial_b\). The Siegel upper half-space \(\Omega=\{z\in\mathbb{C}^n:\mathrm{Im}\,z_n>|z'|^2\}\) has the boundary \(\partial\Omega=\mathbb{H}^{2n+1}\) (the Heisenberg group, with the CR structure from the complex tangent directions, the strictly pseudoconvex Levi form). The \(\bar\partial_b\) on \(\mathbb{H}\): the Folland–Stein / Kohn Laplacian \(\Box_b\) is the sub-Laplacian (the subelliptic operator, the non-elliptic but hypoelliptic, the fundamental solution the Korányi kernel \(|z|^{-2n}\)), and the \(\bar\partial_b\)-problem on \(\mathbb{H}\) is solvable with the subelliptic estimates (the strict pseudoconvexity / the type 2). The Lewy operator \(L=\partial_x+iy\partial_z\) on \(\mathbb{H}\) (the non-solvability: \(L\) is not solvable for some smooth \(f\) — the non-hypoellipticity, the Hansson–Lewy example, the first non-solvable PDE). The key point: the Heisenberg group is the model strictly pseudoconvex CR, the Folland–Stein operators are the subelliptic \(\bar\partial_b\) (the Korányi kernel, the subelliptic estimates), and the Hansson–Lewy non-solvability is the obstruction (the non-pseudoconvex / the non-solvability of the CR PDE).
Check¶
Show that the Heisenberg group \(\mathbb{H}^{2n+1}\) is strictly pseudoconvex (the Levi form positive), and identify the Folland–Stein sub-Laplacian.
The Heisenberg group \(\mathbb{H}^{2n+1}=\{(z,t)\in\mathbb{C}^n\times\mathbb{R}\}\) with the CR structure from the complex tangent directions \(T^{0,1}=\mathrm{span}\{\bar Z_j\}\), \(Z_j=\partial_{z_j}+i\bar z_j\partial_t\) (the Lewy vector fields). The Levi form \(\mathcal{L}(\bar Z_j, Z_k)=\delta_{jk}\) (the standard hermitian form, positive definite — the strictly pseudoconvex). The Folland–Stein sub-Laplacian \(\Delta_b=\sum Z_j\bar Z_j+\bar Z_jZ_j\) (the sum of the Lewy fields, the sub-Laplacian), the non-elliptic but hypoelliptic operator with the fundamental solution \(|z|^{-2n}\) (the Korányi kernel, the subelliptic estimates with the \(\varepsilon=1/2\) gain, the strict pseudoconvexity).
Explain why the Lewy operator \(L=\partial_x+iy\partial_z\) is non-solvable for some smooth \(f\), and identify the role of the pseudoconvexity.
The Lewy operator \(L\) on \(\mathbb{H}\) (the Lewy vector field, the \(\bar\partial_b\)-operator on the Heisenberg) is non-solvable: there exists a smooth \(f\) such that \(Lu=f\) has no smooth solution \(u\) (the Hansson–Lewy example, the first non-solvable PDE / the non-hypoellipticity). The reason: the Lewy operator is the \(\bar\partial_b\) on the non-pseudoconvex side (the Siegel domain is pseudoconvex on one side, non-pseudoconvex on the other — the \(L\) is the \(\bar\partial_b\) for the non-pseudoconvex side, where the solvability fails). The role of the pseudoconvexity: the strict pseudoconvexity (the positive Levi form) gives the subelliptic estimates / the solvability (the Folland–Stein, the Korányi kernel); the non-pseudoconvex (the negative / the degenerate Levi form) gives the non-solvability (the Lewy). The pseudoconvexity is the obstruction / the criterion for the \(\bar\partial_b\) solvability.
Pitfalls¶
- The Lewy non-solvability is for the non-pseudoconvex CR — the strict pseudoconvexity gives the solvability (the Folland–Stein, the subelliptic estimates), the non-pseudoconvex the non-solvability (the Hansson–Lewy). Do not conflate the two.
- The CR embedding (Boutet de Monvel) is for dimension \(\ge 5\) — the dimension-3 CR / the non-strict may not embed (the Rossi example, the non-embeddable CR structures). The dimension \(\ge 5\) is the hypothesis.
- The Tanaka–Webster connection is the canonical connection on the strictly pseudoconvex CR (the pseudo-hermitian geometry), not the Levi-Civita (the CR is not Riemannian, the pseudo-hermitian / the contact structure).
- The Folland–Stein operators are subelliptic (not elliptic — the sub-Laplacian misses the vertical direction, the non-elliptic but hypoelliptic via the Hörmander bracket-generating condition).
- The Heisenberg group is the model strictly pseudoconvex CR (the boundary of the Siegel domain), but the general CR geometry is more complex (the non-Heisenberg CR, the non-homogeneous, the non-strict).
- The Kohn–Rossi cohomology \(H^q_b\) is the \(\bar\partial_b\)-cohomology (the CR cohomology), not the de Rham / the Dolbeault cohomology — the tangential / the CR-specific.
Refs¶
- Kohn–Rossi, "On the extension of holomorphic functions on the boundary of a manifold" (Annals 1965)
- Lewy, "An example of a smooth linear PDE without solution" (Annals 1957); Hansson–Wermer (the CR extension)
- Boutet de Monvel, "Embedding of compact CR manifolds" (1975); Rossi, "Attaching analytic spaces to an analytic space" (1965)
- Folland–Stein, "Estimates for the \(\bar\partial_b\)-complex and analysis on the Heisenberg group" (CPAM 1974); Tanaka, "A differential geometric study on strongly pseudo-convex manifolds" (1975); Korányi–Reimann, "Quasiconformal mappings on the Heisenberg group"
Next¶
SCV index · Foundations · \(\bar\partial\)-Neumann & Kohn Estimates · Stein & Oka Principle · Analytic Methods & \(L^2\) · CG / Legendrian & Transverse — the Legendrian / the CR / the strictly pseudoconvex · Complex geometry · Functional analysis