Geometric properties of non-compact \(CR\) manifolds.

*(English)*Zbl 1200.32022
Tesi. Scuola Normale Superiore Pisa (Nuova Serie) 14. Pisa: Edizioni della Normale; Pisa: Scuola Normale Superiore (Diss. 2007) (ISBN 978-88-7642-348-2/pbk). xv, 103 p. (2009).

This monograph represents the author’s 2007 thesis on real submanifolds of \({\mathbb C}^n\). After a concise summary of CR geometry and extension problems for CR functions, Chapter 1 introduces a Lupacciolu-type condition (*) (see the paper) on a non-compact submanifold \(M\), namely, that its closure in \({\mathbb C}P^n\) be disjoint from some complex projective hypersurface.

Chapters 2 and 3 consider the boundary problem: given \(M\) contained in the boundary of an open set \(\Omega\subseteq{\mathbb C}^n\), find conditions on \(n\), \(M\) (such as (*) and being maximally complex), and \(\Omega\) (such as weak pseudoconvexity) that imply \(M\) is the boundary of a complex variety in \(\Omega\). Results of Harvey and Lawson and a slicing argument are used to show the existence of global or “semi-local” solutions of boundary problems in cases where \(\Omega\) is unbounded and \(M\) is closed but non-compact. These chapters are based on a joint work of the author and A. Saracco (the material of Chapter 2 also appeared elsewhere [Int. J. Math. 18, No. 2, 203–218 (2007; Zbl 1140.32025)]).

Chapter 4 starts by reviewing the work of P. Dolbeault, G. Tomassini and D. Zaitsev [C. R., Math., Acad. Sci. Paris 341, No. 6, 343–348 (2005; Zbl 1085.32019)] on the problem of whether a codimension \(2\) real submanifold \(M\) in \({\mathbb C}^n\) is the boundary of a Levi flat hypersurface. Again, the author considers conditions on \(M\), such as (*) and flat ellipticity of CR singularities, applied to some new cases of this boundary problem.

Chapter 5 is a review of analytic multifunctions based on notions going back to Oka. Multifunctions and results of N. V. Shcherbina [Indiana Univ. Math. J. 42, No. 2, 477–503 (1993; Zbl 0798.32026)] are then applied in Chapters 6 and 7 to prove Liouville theorems for foliations: roughly, if a smooth, Levi flat, closed but non-compact submanifold is bounded in certain directions, then the foliating complex subvarieties are affine subspaces.

Chapters 2 and 3 consider the boundary problem: given \(M\) contained in the boundary of an open set \(\Omega\subseteq{\mathbb C}^n\), find conditions on \(n\), \(M\) (such as (*) and being maximally complex), and \(\Omega\) (such as weak pseudoconvexity) that imply \(M\) is the boundary of a complex variety in \(\Omega\). Results of Harvey and Lawson and a slicing argument are used to show the existence of global or “semi-local” solutions of boundary problems in cases where \(\Omega\) is unbounded and \(M\) is closed but non-compact. These chapters are based on a joint work of the author and A. Saracco (the material of Chapter 2 also appeared elsewhere [Int. J. Math. 18, No. 2, 203–218 (2007; Zbl 1140.32025)]).

Chapter 4 starts by reviewing the work of P. Dolbeault, G. Tomassini and D. Zaitsev [C. R., Math., Acad. Sci. Paris 341, No. 6, 343–348 (2005; Zbl 1085.32019)] on the problem of whether a codimension \(2\) real submanifold \(M\) in \({\mathbb C}^n\) is the boundary of a Levi flat hypersurface. Again, the author considers conditions on \(M\), such as (*) and flat ellipticity of CR singularities, applied to some new cases of this boundary problem.

Chapter 5 is a review of analytic multifunctions based on notions going back to Oka. Multifunctions and results of N. V. Shcherbina [Indiana Univ. Math. J. 42, No. 2, 477–503 (1993; Zbl 0798.32026)] are then applied in Chapters 6 and 7 to prove Liouville theorems for foliations: roughly, if a smooth, Levi flat, closed but non-compact submanifold is bounded in certain directions, then the foliating complex subvarieties are affine subspaces.

Reviewer: Adam Coffman (Fort Wayne)