Estimates for the \(\bar\partial\)-Neumann problem and nonexistence of \(C^2\) Levi-flat hypersurfaces in \(\mathbb{C} P^n\). (English) Zbl 1057.32012
The article contains two main results, both of which are significant.
Theorem 1: There exists no real Levi-flat hypersurface of class \(C^2\) in complex projective space \({\mathbb C} P^n\) (\(n\geq2\)). Theorem 2: If \(\Omega\) is a pseudoconvex domain with class \(C^2\) boundary in \({\mathbb C}P^n\) (\(n\geq2\)), then the \(\overline{\partial}\)-Neumann operator \(N\) exists on square-integrable \((p,q)\)-forms; moreover, there exists a positive \(\epsilon\) (depending on the order of plurisubharmonicity of the distance function of \(\Omega\)) such that \(N\), the operator \(\overline{\partial} N\), the operator \(\overline{\partial}^{*} N\), and the Bergman projection are continuous on the Sobolev space \(W^s(\Omega)\) when \(0<s<\epsilon\).
Y.-T. Siu previously proved theorem 1 under a stronger smoothness hypothesis, first when \(n\geq3\) [Ann. Math. (2) 151, No. 3, 1217–1243 (2000; Zbl 0980.53065)] and then when \(n=2\) [Ann. Math. (2) 156, No. 2, 595–621 (2002; Zbl 1030.53071)]. The real-analytic case was proved when \(n\geq3\) by A. Lins Neto [Ann. Inst. Fourier 49, No. 4, 1369–1385 (1999; Zbl 0963.32022)] and when \(n=2\) by T. Ohsawa [Nagoya Math. J. 158, 95–98 (2000; Zbl 1023.32026)].
The authors offer two proofs of theorem 1, both based on theorem 2, which is itself a fundamental result of independent interest. The first proof, based on solving a \(\overline{\partial}_b\)-problem by extending to the interior, works when \(n\geq3\) and the boundary is smooth of class \(C^{2,\alpha}\) for some positive \(\alpha\).
The second proof is based on a Liouville-type theorem that the authors state as
Proposition 4.5: If \(\Omega\) is a pseudoconcave domain in \({\mathbb C} P^n\) (\(n\geq2\)) with class \(C^2\) boundary, then the space of \(\overline{\partial}\)-closed square-integrable \((p,0)\)-forms on \(\Omega\) is trivial when \(1\leq p\leq n\) and consists of constant functions when \(p=0\).
The erratum attempts to fix a gap in the proof of this proposition, but in a subsequent paper [\(\overline{\partial}\)-closed extensions of forms and nonexistence of Levi-flat hypersurfaces in \({\mathbb C} P^2\), preprint (2004)] the authors reformulate the proposition as a conjecture, stating that it remains unproved. That paper explains how to modify the first proof of theorem 1 to make the proof work for class \(C^2\) boundary when \(n\geq2\).
Theorem 1: There exists no real Levi-flat hypersurface of class \(C^2\) in complex projective space \({\mathbb C} P^n\) (\(n\geq2\)). Theorem 2: If \(\Omega\) is a pseudoconvex domain with class \(C^2\) boundary in \({\mathbb C}P^n\) (\(n\geq2\)), then the \(\overline{\partial}\)-Neumann operator \(N\) exists on square-integrable \((p,q)\)-forms; moreover, there exists a positive \(\epsilon\) (depending on the order of plurisubharmonicity of the distance function of \(\Omega\)) such that \(N\), the operator \(\overline{\partial} N\), the operator \(\overline{\partial}^{*} N\), and the Bergman projection are continuous on the Sobolev space \(W^s(\Omega)\) when \(0<s<\epsilon\).
Y.-T. Siu previously proved theorem 1 under a stronger smoothness hypothesis, first when \(n\geq3\) [Ann. Math. (2) 151, No. 3, 1217–1243 (2000; Zbl 0980.53065)] and then when \(n=2\) [Ann. Math. (2) 156, No. 2, 595–621 (2002; Zbl 1030.53071)]. The real-analytic case was proved when \(n\geq3\) by A. Lins Neto [Ann. Inst. Fourier 49, No. 4, 1369–1385 (1999; Zbl 0963.32022)] and when \(n=2\) by T. Ohsawa [Nagoya Math. J. 158, 95–98 (2000; Zbl 1023.32026)].
The authors offer two proofs of theorem 1, both based on theorem 2, which is itself a fundamental result of independent interest. The first proof, based on solving a \(\overline{\partial}_b\)-problem by extending to the interior, works when \(n\geq3\) and the boundary is smooth of class \(C^{2,\alpha}\) for some positive \(\alpha\).
The second proof is based on a Liouville-type theorem that the authors state as
Proposition 4.5: If \(\Omega\) is a pseudoconcave domain in \({\mathbb C} P^n\) (\(n\geq2\)) with class \(C^2\) boundary, then the space of \(\overline{\partial}\)-closed square-integrable \((p,0)\)-forms on \(\Omega\) is trivial when \(1\leq p\leq n\) and consists of constant functions when \(p=0\).
The erratum attempts to fix a gap in the proof of this proposition, but in a subsequent paper [\(\overline{\partial}\)-closed extensions of forms and nonexistence of Levi-flat hypersurfaces in \({\mathbb C} P^2\), preprint (2004)] the authors reformulate the proposition as a conjecture, stating that it remains unproved. That paper explains how to modify the first proof of theorem 1 to make the proof work for class \(C^2\) boundary when \(n\geq2\).
Reviewer: Harold P. Boas (College Station)
MSC:
| 32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |
| 32V40 | Real submanifolds in complex manifolds |
| 35N15 | \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs |
| 53C40 | Global submanifolds |
Keywords:
projective space; tangential Cauchy-Riemann equation; pseudoconvex domain; pseudoconcave domain; Bergman projection; Sobolev space; Liouville theorem; plurisubharmonicityReferences:
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